Business Statistics, 4th by Ken Black Chapter 1 Introduction to Statistics Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Define statistics Become aware of a wide range of applications of statistics in business Differentiate between descriptive and inferential statistics Classify numbers by level of data and understand why doing so is important Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2 2

Statistics in Business Accounting — auditing and cost estimation Economics — regional, national, and international economic performance Finance — investments and portfolio management Management — human resources, compensation, and quality management Management Information Systems — performance of systems which gather, summarize, and disseminate information to various managerial levels Marketing — market analysis and consumer research International Business — market and demographic analysis Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7 8

What is Statistics? Science of gathering, analyzing, interpreting, and presenting data Branch of mathematics Course of study Facts and figures A death Measurement taken on a sample Type of distribution being used to analyze data Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8 11

Population Versus Sample Population — the whole a collection of persons, objects, or items under study Census — gathering data from the entire population Sample — a portion of the whole a subset of the population Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9 12

Population Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Population and Census Data Identifier Color MPG RD1 Red 12 RD2 10 RD3 13 RD4 RD5 BL1 Blue 27 BL2 24 GR1 Green 35 GR2 GY1 Gray 15 GY2 18 GY3 17 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11 14

Sample and Sample Data Identifier Color MPG RD2 Red 10 RD5 13 GR1 Green 35 GY2 Gray 18 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Descriptive vs. Inferential Statistics Descriptive Statistics — using data gathered on a group to describe or reach conclusions about that same group only Inferential Statistics — using sample data to reach conclusions about the population from which the sample was taken Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13 16

Parameter vs. Statistic Parameter — descriptive measure of the population Usually represented by Greek letters Statistic — descriptive measure of a sample Usually represented by Roman letters Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14 17

Symbols for Population Parameters Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15 18

Symbols for Sample Statistics Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16 19

Process of Inferential Statistics Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17 20

Levels of Data Measurement Nominal — Lowest level of measurement Ordinal Interval Ratio — Highest level of measurement Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18 21

Nominal Level Data Numbers are used to classify or categorize Example: Employment Classification 1 for Educator 2 for Construction Worker 3 for Manufacturing Worker Example: Ethnicity 1 for African-American 2 for Anglo-American 3 for Hispanic-American Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19 22

Ordinal Level Data Numbers are used to indicate rank or order Relative magnitude of numbers is meaningful Differences between numbers are not comparable Example: Ranking productivity of employees Example: Taste test ranking of three brands of soft drink Example: Position within an organization 1 for President 2 for Vice President 3 for Plant Manager 4 for Department Supervisor 5 for Employee Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20 23

Example of Ordinal Measurement Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21 24

Ordinal Data Faculty and staff should receive preferential treatment for parking space. 1 2 3 4 5 Strongly Agree Disagree Neutral Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Interval Level Data Distances between consecutive integers are equal Relative magnitude of numbers is meaningful Differences between numbers are comparable Location of origin, zero, is arbitrary Vertical intercept of unit of measure transform function is not zero Example: Fahrenheit Temperature Example: Calendar Time Example: Monetary Utility Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22 26

Ratio Level Data Highest level of measurement Relative magnitude of numbers is meaningful Differences between numbers are comparable Location of origin, zero, is absolute (natural) Vertical intercept of unit of measure transform function is zero Examples: Height, Weight, and Volume Example: Monetary Variables, such as Profit and Loss, Revenues, and Expenses Example: Financial ratios, such as P/E Ratio, Inventory Turnover, and Quick Ratio. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23 27

Usage Potential of Various Levels of Data Ratio Interval Ordinal Nominal Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24 28

Data Level, Operations, and Statistical Methods Nominal Ordinal Interval Ratio Meaningful Operations Classifying and Counting All of the above plus Ranking All of the above plus Addition, Subtraction, Multiplication, and Division All of the above Statistical Methods Nonparametric Parametric Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25 29

Business Statistics, 4e by Ken Black Chapter 2 Charts & Graphs Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Recognize the difference between grouped and ungrouped data Construct a frequency distribution Construct a histogram, a frequency polygon, an ogive, a pie chart, a stem and leaf plot, a Pareto chart, and a scatter plot Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Ungrouped Versus Grouped Data Ungrouped data have not been summarized in any way are also called raw data Grouped data have been organized into a frequency distribution Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Example of Ungrouped Data 42 30 53 50 52 55 49 61 74 26 58 40 28 36 33 31 37 32 23 43 29 34 47 35 64 46 57 25 60 54 Ages of a Sample of Managers from Urban Child Care Centers in the United States Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Frequency Distribution of Child Care Manager’s Ages Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 50-under 60 11 60-under 70 3 70-under 80 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Data Range Smallest Largest 42 30 53 50 52 55 49 61 74 26 58 40 28 36 33 31 37 32 23 43 29 34 47 35 64 46 57 25 60 54 Smallest Largest Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Number of Classes and Class Width The number of classes should be between 5 and 15. Fewer than 5 classes cause excessive summarization. More than 15 classes leave too much detail. Class Width Divide the range by the number of classes for an approximate class width Round up to a convenient number Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Class Midpoint Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Relative Frequency Relative Class Interval Frequency Frequency 20-under 30 6 .12 30-under 40 18 .36 40-under 50 11 .22 50-under 60 11 .22 60-under 70 3 .06 70-under 80 1 .02 Total 50 1.00 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Cumulative Frequency Cumulative Class Interval Frequency Frequency 20-under 30 6 6 30-under 40 18 24 40-under 50 11 35 50-under 60 11 46 60-under 70 3 49 70-under 80 1 50 Total 50 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Class Midpoints, Relative Frequencies, and Cumulative Frequencies Relative Cumulative Class Interval Frequency Midpoint Frequency Frequency 20-under 30 6 25 .12 6 30-under 40 18 35 .36 24 40-under 50 11 45 .22 35 50-under 60 11 55 .22 46 60-under 70 3 65 .06 49 70-under 80 1 75 .02 50 Total 50 1.00 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Cumulative Relative Frequencies Relative Cumulative Relative Class Interval Frequency Frequency Frequency Frequency 20-under 30 6 .12 6 .12 30-under 40 18 .36 24 .48 40-under 50 11 .22 35 .70 50-under 60 11 .22 46 .92 60-under 70 3 .06 49 .98 70-under 80 1 .02 50 1.00 Total 50 1.00 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Common Statistical Graphs Histogram -- vertical bar chart of frequencies Frequency Polygon -- line graph of frequencies Ogive -- line graph of cumulative frequencies Pie Chart -- proportional representation for categories of a whole Stem and Leaf Plot Pareto Chart Scatter Plot Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 Histogram Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 50-under 60 11 60-under 70 3 70-under 80 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Histogram Construction Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 50-under 60 11 60-under 70 3 70-under 80 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 Frequency Polygon Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 50-under 60 11 60-under 70 3 70-under 80 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Ogive Cumulative Class Interval Frequency 20-under 30 6 30-under 40 24 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Relative Frequency Ogive Cumulative Relative Class Interval Frequency 20-under 30 .12 30-under 40 .48 40-under 50 .70 50-under 60 .92 60-under 70 .98 70-under 80 1.00 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

Complaints by Amtrak Passengers NUMBER PROPORTION DEGREES Stations, etc. 28,000 .40 144.0 Train Performance 14,700 .21 75.6 Equipment 10,500 .15 50.4 Personnel 9,800 .14 50.6 Schedules, etc. 7,000 .10 36.0 Total 70,000 1.00 360.0 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Complaints by Amtrak Passengers Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Second Quarter Truck Production in the U.S. (Hypothetical values) Company A B C D E Totals 357,411 354,936 160,997 34,099 12,747 920,190 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Second Quarter U.S. Truck Production Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Pie Chart Calculations for Company A 2d Quarter Truck Production Proportion Degrees Company A B C D E Totals 357,411 354,936 160,997 34,099 12,747 920,190 .388 .386 .175 .037 .014 1.000 140 139 63 13 5 360 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Safety Examination Scores for Plant Trainees Raw Data Stem Leaf 2 3 4 5 6 7 8 9 3 9 7 9 5 6 9 0 7 7 8 8 0 2 4 5 5 6 7 7 8 9 1 1 2 3 3 6 8 9 1 1 2 4 7 86 76 23 77 81 79 68 92 59 75 83 49 91 47 72 82 74 70 56 60 88 97 39 78 94 55 67 89 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Construction of Stem and Leaf Plot Raw Data Stem Leaf 2 3 4 5 6 7 8 9 3 9 7 9 5 6 9 0 7 7 8 8 0 2 4 5 5 6 7 7 8 9 1 1 2 3 3 6 8 9 1 1 2 4 7 86 76 23 77 81 79 68 92 59 75 83 49 91 47 72 82 74 70 56 60 88 97 39 78 94 55 67 89 Stem Leaf Stem Leaf Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Pareto Chart Frequency 10 20 30 40 50 60 70 80 90 100 Poor Wiring 10 20 30 40 50 60 70 80 90 100 Poor Wiring Short in Coil Defective Plug Other Frequency 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Scatter Plot Registered Vehicles (1000's) Gasoline Sales (1000's of Gallons) 5 60 15 120 9 90 140 7 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Business Statistics, 4e by Ken Black Chapter 3 Descriptive Statistics Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Distinguish between measures of central tendency, measures of variability, measures of shape, and measures of association. Understand the meanings of mean, median, mode, quartile, percentile, and range. Compute mean, median, mode, percentile, quartile, range, variance, standard deviation, and mean absolute deviation on ungrouped data. Differentiate between sample and population variance and standard deviation. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Learning Objectives -- Continued Understand the meaning of standard deviation as it is applied by using the empirical rule and Chebyshev’s theorem. Compute the mean, median, standard deviation, and variance on grouped data. Understand box and whisker plots, skewness, and kurtosis. Compute a coefficient of correlation and interpret it. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 3

Measures of Central Tendency: Ungrouped Data Measures of central tendency yield information about “particular places or locations in a group of numbers.” Common Measures of Location Mode Median Mean Percentiles Quartiles Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Mode The most frequently occurring value in a data set Applicable to all levels of data measurement (nominal, ordinal, interval, and ratio) Bimodal -- Data sets that have two modes Multimodal -- Data sets that contain more than two modes Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Mode -- Example The mode is 44. There are more 44s than any other value. 35 37 39 40 41 43 44 45 46 48 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Median Middle value in an ordered array of numbers. Applicable for ordinal, interval, and ratio data Not applicable for nominal data Unaffected by extremely large and extremely small values. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Median: Computational Procedure First Procedure Arrange the observations in an ordered array. If there is an odd number of terms, the median is the middle term of the ordered array. If there is an even number of terms, the median is the average of the middle two terms. Second Procedure The median’s position in an ordered array is given by (n+1)/2. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Median: Example with an Odd Number of Terms Ordered Array 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22 There are 17 terms in the ordered array. Position of median = (n+1)/2 = (17+1)/2 = 9 The median is the 9th term, 15. If the 22 is replaced by 100, the median is 15. If the 3 is replaced by -103, the median is 15. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

with an Even Number of Terms Median: Example with an Even Number of Terms Ordered Array 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 There are 16 terms in the ordered array. Position of median = (n+1)/2 = (16+1)/2 = 8.5 The median is between the 8th and 9th terms, 14.5. If the 21 is replaced by 100, the median is 14.5. If the 3 is replaced by -88, the median is 14.5. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Arithmetic Mean Commonly called ‘the mean’ is the average of a group of numbers Applicable for interval and ratio data Not applicable for nominal or ordinal data Affected by each value in the data set, including extreme values Computed by summing all values in the data set and dividing the sum by the number of values in the data set Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Population Mean Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Sample Mean Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Percentiles Measures of central tendency that divide a group of data into 100 parts At least n% of the data lie below the nth percentile, and at most (100 - n)% of the data lie above the nth percentile Example: 90th percentile indicates that at least 90% of the data lie below it, and at most 10% of the data lie above it The median and the 50th percentile have the same value. Applicable for ordinal, interval, and ratio data Not applicable for nominal data Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Percentiles: Computational Procedure Organize the data into an ascending ordered array. Calculate the percentile location: Determine the percentile’s location and its value. If i is a whole number, the percentile is the average of the values at the i and (i+1) positions. If i is not a whole number, the percentile is at the (i+1) position in the ordered array. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Percentiles: Example Raw Data: 14, 12, 19, 23, 5, 13, 28, 17 Ordered Array: 5, 12, 13, 14, 17, 19, 23, 28 Location of 30th percentile: The location index, i, is not a whole number; i+1 = 2.4+1=3.4; the whole number portion is 3; the 30th percentile is at the 3rd location of the array; the 30th percentile is 13. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Quartiles Measures of central tendency that divide a group of data into four subgroups Q1: 25% of the data set is below the first quartile Q2: 50% of the data set is below the second quartile Q3: 75% of the data set is below the third quartile Q1 is equal to the 25th percentile Q2 is located at 50th percentile and equals the median Q3 is equal to the 75th percentile Quartile values are not necessarily members of the data set Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Quartiles 25% Q3 Q2 Q1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Quartiles: Example Ordered array: 106, 109, 114, 116, 121, 122, 125, 129 Q1 Q2: Q3: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Variability No Variability in Cash Flow Variability in Cash Flow Mean Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Variability Variability No Variability 21 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Measures of Variability: Ungrouped Data Measures of variability describe the spread or the dispersion of a set of data. Common Measures of Variability Range Interquartile Range Mean Absolute Deviation Variance Standard Deviation Z scores Coefficient of Variation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

Range The difference between the largest and the smallest values in a set of data Simple to compute Ignores all data points except the two extremes Example: Range = Largest - Smallest = 48 - 35 = 13 35 37 39 40 41 43 44 45 46 48 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

Interquartile Range Range of values between the first and third quartiles Range of the “middle half” Less influenced by extremes Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 37

Deviation from the Mean Data set: 5, 9, 16, 17, 18 Mean: Deviations from the mean: -8, -4, 3, 4, 5 -4 +5 -8 +4 +3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

Mean Absolute Deviation Average of the absolute deviations from the mean 5 9 16 17 18 -8 -4 +3 +4 +5 +8 24 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Population Variance 5 9 16 17 18 -8 -4 +3 +4 +5 64 25 130 Average of the squared deviations from the arithmetic mean 5 9 16 17 18 -8 -4 +3 +4 +5 64 25 130 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Population Standard Deviation Square root of the variance 5 9 16 17 18 -8 -4 +3 +4 +5 64 25 130 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Sample Variance Average of the squared deviations from the arithmetic mean 2,398 1,844 1,539 1,311 7,092 625 71 -234 -462 390,625 5,041 54,756 213,444 663,866 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Sample Standard Deviation 2,398 1,844 1,539 1,311 7,092 625 71 -234 -462 390,625 5,041 54,756 213,444 663,866 Square root of the sample variance Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Uses of Standard Deviation Indicator of financial risk Quality Control construction of quality control charts process capability studies Comparing populations household incomes in two cities employee absenteeism at two plants Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

Standard Deviation as an Indicator of Financial Risk Annualized Rate of Return Financial Security   A 15% 3% B 7% Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Empirical Rule Data are normally distributed (or approximately normal) 95 99.7 68 Distance from the Mean Percentage of Values Falling Within Distance Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Chebyshev’s Theorem Applies to all distributions 33 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 33

Chebyshev’s Theorem 1-1/22 = 0.75 K = 2 1-1/32 = 0.89 K = 3 K = 4 Applies to all distributions 1-1/32 = 0.89 1-1/22 = 0.75 Distance from the Mean Minimum Proportion of Values Falling Within Distance Number of Standard Deviations K = 2 K = 3 K = 4 1-1/42 = 0.94 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

Coefficient of Variation Ratio of the standard deviation to the mean, expressed as a percentage Measurement of relative dispersion Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Coefficient of Variation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 36

Measures of Central Tendency and Variability: Grouped Data Mean Median Mode Measures of Variability Variance Standard Deviation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 38

Mean of Grouped Data Weighted average of class midpoints Class frequencies are the weights Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Calculation of Grouped Mean Class Interval Frequency Class Midpoint fM 20-under 30 6 25 150 30-under 40 18 35 630 40-under 50 11 45 495 50-under 60 11 55 605 60-under 70 3 65 195 70-under 80 1 75 75 50 2150 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Median of Grouped Data Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 41

Median of Grouped Data -- Example Cumulative Class Interval Frequency Frequency 20-under 30 6 6 30-under 40 18 24 40-under 50 11 35 50-under 60 11 46 60-under 70 3 49 70-under 80 1 50 N = 50 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 42

Class Interval Frequency Mode of Grouped Data Midpoint of the modal class Modal class has the greatest frequency Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 50-under 60 11 60-under 70 3 70-under 80 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 43

Variance and Standard Deviation of Grouped Data Population Sample Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 44

Population Variance and Standard Deviation of Grouped Data 6 18 11 3 1 50 25 35 45 55 65 75 150 630 495 605 195 75 2150 -18 -8 2 12 22 32 20-under 30 30-under 40 40-under 50 50-under 60 60-under 70 70-under 80 1944 1152 44 1584 1452 1024 7200 324 64 4 144 484 1024 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 45

Measures of Shape Skewness Kurtosis Box and Whisker Plots Absence of symmetry Extreme values in one side of a distribution Kurtosis Peakedness of a distribution Leptokurtic: high and thin Mesokurtic: normal shape Platykurtic: flat and spread out Box and Whisker Plots Graphic display of a distribution Reveals skewness Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 46

Skewness Negatively Skewed Positively Skewed Symmetric (Not Skewed) 47 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 47

Skewness Negatively Symmetric Positively Skewed (Not Skewed) Mean Mode Median Mean Symmetric (Not Skewed) Positively Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 48

Coefficient of Skewness Summary measure for skewness If S < 0, the distribution is negatively skewed (skewed to the left). If S = 0, the distribution is symmetric (not skewed). If S > 0, the distribution is positively skewed (skewed to the right). Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 49

Coefficient of Skewness Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 50

Kurtosis Peakedness of a distribution Leptokurtic: high and thin Mesokurtic: normal in shape Platykurtic: flat and spread out Leptokurtic Mesokurtic Platykurtic Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 51

Box and Whisker Plot Five secific values are used: Inner Fences Median, Q2 First quartile, Q1 Third quartile, Q3 Minimum value in the data set Maximum value in the data set Inner Fences IQR = Q3 - Q1 Lower inner fence = Q1 - 1.5 IQR Upper inner fence = Q3 + 1.5 IQR Outer Fences Lower outer fence = Q1 - 3.0 IQR Upper outer fence = Q3 + 3.0 IQR Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 52

Box and Whisker Plot Q1 Q3 Q2 Minimum Maximum 53 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 53

Skewness: Box and Whisker Plots, and Coefficient of Skewness Negatively Skewed Positively Symmetric (Not Skewed) S < 0 S = 0 S > 0 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 54

Pearson Product-Moment Correlation Coefficient Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Three Degrees of Correlation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Computation of r for the Economics Example (Part 1) Day Interest X Futures Index Y 1 7.43 221 55.205 48,841 1,642.03 2 7.48 222 55.950 49,284 1,660.56 3 8.00 226 64.000 51,076 1,808.00 4 7.75 225 60.063 50,625 1,743.75 5 7.60 224 57.760 50,176 1,702.40 6 7.63 223 58.217 49,729 1,701.49 7 7.68 58.982 1,712.64 8 7.67 58.829 1,733.42 9 7.59 57.608 1,715.34 10 8.07 235 65.125 55,225 1,896.45 11 8.03 233 64.481 54,289 1,870.99 12 241 58,081 1,928.00 Summations 92.93 2,725 720.220 619,207 21,115.07 X2 Y2 XY Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 41

Computation of r for the Economics Example (Part 2) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 43

Scatter Plot and Correlation Matrix for the Economics Example 220 225 230 235 240 245 7.40 7.60 7.80 8.00 8.20 Interest Futures Index Interest Futures Index 1 0.815254 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Business Statistics, 4e by Ken Black Chapter 4 Probability Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Comprehend the different ways of assigning probability. Understand and apply marginal, union, joint, and conditional probabilities. Select the appropriate law of probability to use in solving problems. Solve problems using the laws of probability including the laws of addition, multiplication and conditional probability Revise probabilities using Bayes’ rule. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Methods of Assigning Probabilities Classical method of assigning probability (rules and laws) Relative frequency of occurrence (cumulated historical data) Subjective Probability (personal intuition or reasoning) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 3

Classical Probability Number of outcomes leading to the event divided by the total number of outcomes possible Each outcome is equally likely Determined a priori -- before performing the experiment Applicable to games of chance Objective -- everyone correctly using the method assigns an identical probability Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Relative Frequency Probability Based on historical data Computed after performing the experiment Number of times an event occurred divided by the number of trials Objective -- everyone correctly using the method assigns an identical probability Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Subjective Probability Comes from a person’s intuition or reasoning Subjective -- different individuals may (correctly) assign different numeric probabilities to the same event Degree of belief Useful for unique (single-trial) experiments New product introduction Initial public offering of common stock Site selection decisions Sporting events Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Structure of Probability Experiment Event Elementary Events Sample Space Unions and Intersections Mutually Exclusive Events Independent Events Collectively Exhaustive Events Complementary Events Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Experiment Experiment: a process that produces outcomes More than one possible outcome Only one outcome per trial Trial: one repetition of the process Elementary Event: cannot be decomposed or broken down into other events Event: an outcome of an experiment may be an elementary event, or may be an aggregate of elementary events usually represented by an uppercase letter, e.g., A, E1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

An Example Experiment Tiny Town Population Experiment: randomly select, without replacement, two families from the residents of Tiny Town Family Children in Household Number of Automobiles A B C D Yes No 3 2 1 Tiny Town Population Elementary Event: the sample includes families A and C Event: each family in the sample has children in the household Event: the sample families own a total of four automobiles Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Sample Space The set of all elementary events for an experiment Methods for describing a sample space roster or listing tree diagram set builder notation Venn diagram Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Sample Space: Roster Example Experiment: randomly select, without replacement, two families from the residents of Tiny Town Each ordered pair in the sample space is an elementary event, for example -- (D,C) Family Children in Household Number of Automobiles A B C D Yes No 3 2 1 Listing of Sample Space (A,B), (A,C), (A,D), (B,A), (B,C), (B,D), (C,A), (C,B), (C,D), (D,A), (D,B), (D,C) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Sample Space: Tree Diagram for Random Sample of Two Families B C D Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Sample Space: Set Notation for Random Sample of Two Families S = {(x,y) | x is the family selected on the first draw, and y is the family selected on the second draw} Concise description of large sample spaces Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Listing of Sample Space Useful for discussion of general principles and concepts Listing of Sample Space (A,B), (A,C), (A,D), (B,A), (B,C), (B,D), (C,A), (C,B), (C,D), (D,A), (D,B), (D,C) Venn Diagram Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Union of Sets The union of two sets contains an instance of each element of the two sets. Y X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Intersection of Sets The intersection of two sets contains only those element common to the two sets. Y X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Mutually Exclusive Events Events with no common outcomes Occurrence of one event precludes the occurrence of the other event Y X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Independent Events Occurrence of one event does not affect the occurrence or nonoccurrence of the other event The conditional probability of X given Y is equal to the marginal probability of X. The conditional probability of Y given X is equal to the marginal probability of Y. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Collectively Exhaustive Events Contains all elementary events for an experiment E1 E2 E3 Sample Space with three collectively exhaustive events Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Complementary Events All elementary events not in the event ‘A’ are in its complementary event. Sample Space A Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Counting the Possibilities mn Rule Sampling from a Population with Replacement Combinations: Sampling from a Population without Replacement Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

mn Rule If an operation can be done m ways and a second operation can be done n ways, then there are mn ways for the two operations to occur in order. A cafeteria offers 5 salads, 4 meats, 8 vegetables, 3 breads, 4 desserts, and 3 drinks. A meal is two servings of vegetables, which may be identical, and one serving each of the other items. How many meals are available? Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Sampling from a Population with Replacement A tray contains 1,000 individual tax returns. If 3 returns are randomly selected with replacement from the tray, how many possible samples are there? (N)n = (1,000)3 = 1,000,000,000 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Combinations A tray contains 1,000 individual tax returns. If 3 returns are randomly selected without replacement from the tray, how many possible samples are there? Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Four Types of Probability Marginal Probability Union Probability Joint Probability Conditional Probability Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Four Types of Probability Marginal The probability of X occurring Union The probability of X or Y occurring Joint The probability of X and Y occurring Conditional The probability of X occurring given that Y has occurred Y X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

General Law of Addition Y X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

General Law of Addition -- Example S N .56 .67 .70 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

Office Design Problem Probability Matrix .11 .19 .30 .56 .14 .70 .67 .33 1.00 Increase Storage Space Yes No Total Noise Reduction Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Office Design Problem Probability Matrix .11 .19 .30 .56 .14 .70 .67 .33 1.00 Increase Storage Space Yes No Total Noise Reduction Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Office Design Problem Probability Matrix .11 .19 .30 .56 .14 .70 .67 .33 1.00 Increase Storage Space Yes No Total Noise Reduction Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Venn Diagram of the X or Y but not Both Case Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

The Neither/Nor Region Y X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

The Neither/Nor Region S N Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Special Law of Addition X Y Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Demonstration Problem 4.3 Type of Gender Position Male Female Total Managerial 8 3 11 Professional 31 13 44 Technical 52 17 69 Clerical 9 22 100 55 155 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Demonstration Problem 4.3 Type of Gender Position Male Female Total Managerial 8 3 11 Professional 31 13 44 Technical 52 17 69 Clerical 9 22 100 55 155 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Law of Multiplication Demonstration Problem 4.5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

Law of Multiplication Demonstration Problem 4.5 Total .7857 Yes No .4571 .3286 .1143 .1000 .2143 .5714 .4286 1.00 Married Supervisor Probability Matrix of Employees Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Special Law of Multiplication for Independent Events General Law Special Law Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 36

Law of Conditional Probability The conditional probability of X given Y is the joint probability of X and Y divided by the marginal probability of Y. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 37

Law of Conditional Probability S .56 .70 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 38

Reduced Sample Space for “Increase Storage Space” = “Yes” Office Design Problem .19 .30 .14 .70 .33 1.00 Increase Storage Space Yes No Total Noise Reduction .11 .56 .67 Reduced Sample Space for “Increase Storage Space” = “Yes” Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Independent Events If X and Y are independent events, the occurrence of Y does not affect the probability of X occurring. If X and Y are independent events, the occurrence of X does not affect the probability of Y occurring. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Independent Events Demonstration Problem 4.10 Geographic Location Northeast D Southeast E Midwest F West G Finance A .12 .05 .04 .07 .28 Manufacturing B .15 .03 .11 .06 .35 Communications C .14 .09 .08 .37 .41 .17 .21 1.00 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Independent Events Demonstration Problem 4.11 8 12 20 B 30 50 C 6 9 15 34 51 85 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Revision of Probabilities: Bayes’ Rule An extension to the conditional law of probabilities Enables revision of original probabilities with new information Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 43

Revision of Probabilities with Bayes' Rule: Ribbon Problem Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 44

Revision of Probabilities with Bayes’ Rule: Ribbon Problem Conditional Probability 0.052 0.042 0.094 0.65 0.35 0.08 0.12 =0.553 =0.447 Alamo South Jersey Event Prior Probability Joint Probability P E d i ( )  Revised Probability | Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 45

Revision of Probabilities with Bayes' Rule: Ribbon Problem Alamo 0.65 South Jersey 0.35 Defective 0.08 0.12 Acceptable 0.92 0.88 0.052 0.042 + 0.094 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 46

Probability for a Sequence of Independent Trials 25 percent of a bank’s customers are commercial (C) and 75 percent are retail (R). Experiment: Record the category (C or R) for each of the next three customers arriving at the bank. Sequences with 1 commercial and 2 retail customers. C1 R2 R3 R1 C2 R3 R1 R2 C3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 47

Probability for a Sequence of Independent Trials Probability of specific sequences containing 1 commercial and 2 retail customers, assuming the events C and R are independent Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 48

Probability for a Sequence of Independent Trials Probability of observing a sequence containing 1 commercial and 2 retail customers, assuming the events C and R are independent Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 49

Probability for a Sequence of Independent Trials Probability of a specific sequence with 1 commercial and 2 retail customers, assuming the events C and R are independent Number of sequences containing 1 commercial and 2 retail customers Probability of a sequence containing 1 commercial and 2 retail customers Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 50

Probability for a Sequence of Dependent Trials Twenty percent of a batch of 40 tax returns contain errors. Experiment: Randomly select 4 of the 40 tax returns and record whether each return contains an error (E) or not (N). Outcomes with exactly 2 erroneous tax returns E1 E2 N3 N4 E1 N2 E3 N4 E1 N2 N3 E4 N1 E2 E3 N4 N1 E2 N3 E4 N1 N2 E3 E4 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 51

Probability for a Sequence of Dependent Trials Probability of specific sequences containing 2 erroneous tax returns (three of the six sequences) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 52

Probability for a Sequence of Independent Trials Probability of observing a sequence containing exactly 2 erroneous tax returns Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 53

Probability for a Sequence of Dependent Trials Probability of a specific sequence with exactly 2 erroneous tax returns Number of sequences containing exactly 2 erroneous tax returns Probability of a sequence containing exactly 2 erroneous tax returns Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 54

Business Statistics, 4e by Ken Black Chapter 5 Discrete Distributions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Distinguish between discrete random variables and continuous random variables. Know how to determine the mean and variance of a discrete distribution. Identify the type of statistical experiments that can be described by the binomial distribution, and know how to work such problems. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Learning Objectives -- Continued Decide when to use the Poisson distribution in analyzing statistical experiments, and know how to work such problems. Decide when binomial distribution problems can be approximated by the Poisson distribution, and know how to work such problems. Decide when to use the hypergeometric distribution, and know how to work such problems. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 3

Discrete vs Continuous Distributions Random Variable -- a variable which contains the outcomes of a chance experiment Discrete Random Variable -- the set of all possible values is at most a finite or a countably infinite number of possible values Number of new subscribers to a magazine Number of bad checks received by a restaurant Number of absent employees on a given day Continuous Random Variable -- takes on values at every point over a given interval Current Ratio of a motorcycle distributorship Elapsed time between arrivals of bank customers Percent of the labor force that is unemployed Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Some Special Distributions Discrete binomial Poisson hypergeometric Continuous normal uniform exponential t chi-square F Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Discrete Distribution -- Example 1 2 3 4 5 0.37 0.31 0.18 0.09 0.04 0.01 Number of Crises Probability Distribution of Daily Crises 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 P r o b a i l t y Number of Crises Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Requirements for a Discrete Probability Function Probabilities are between 0 and 1, inclusively Total of all probabilities equals 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Requirements for a Discrete Probability Function -- Examples P(X) -1 1 2 3 .1 .2 .4 1.0 X P(X) -1 1 2 3 -.1 .3 .4 .1 1.0 X P(X) -1 1 2 3 .1 .3 .4 1.2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Mean of a Discrete Distribution X -1 1 2 3 P(X) .1 .2 .4 -.1 .0 .3 1.0 P  ( ) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Variance and Standard Deviation of a Discrete Distribution X -1 1 2 3 P(X) .1 .2 .4 -2   4 .0 1.2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Mean of the Crises Data Example P(X)  .37 .00 1 .31 2 .18 .36 3 .09 .27 4 .04 .16 5 .01 .05 1.15 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 P r o b a i l t y Number of Crises Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Variance and Standard Deviation of Crises Data Example P(X) (X-  ) 2  .37 -1.15 1.32 .49 1 .31 -0.15 0.02 .01 .18 0.85 0.72 .13 3 .09 1.85 3.42 4 .04 2.85 8.12 .32 5 3.85 14.82 .15 1.41 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Binomial Distribution Experiment involves n identical trials Each trial has exactly two possible outcomes: success and failure Each trial is independent of the previous trials p is the probability of a success on any one trial q = (1-p) is the probability of a failure on any one trial p and q are constant throughout the experiment X is the number of successes in the n trials Applications Sampling with replacement Sampling without replacement -- n < 5% N Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Binomial Distribution Probability function Mean value Variance and standard deviation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Binomial Distribution: Development Experiment: randomly select, with replacement, two families from the residents of Tiny Town Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ Family Children in Household Number of Automobiles A B C D Yes No 3 2 1 Listing of Sample Space (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Binomial Distribution: Development Continued (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D) Listing of Sample Space 2 1 X 1/16 P(outcome) Families A, B, and D have children in the household; family C does not Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Binomial Distribution: Development Continued (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D) Listing of Sample Space 2 1 X 1/16 P(outcome) 6/16 9/16 P(X) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Binomial Distribution: Development Continued Families A, B, and D have children in the household; family C does not Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ X Possible Sequences 1 2 (F,F) (S,F) (F,S) (S,S) P(sequence) (. ) ( . 25  75 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Binomial Distribution: Development Continued X Possible Sequences 1 2 (F,F) (S,F) (F,S) (S,S) P(sequence) (. ) ( . 25  75 P(X) =0.375 =0.5625 =0.0625 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Binomial Distribution: Demonstration Problem 5.3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Binomial Table n = 20 PROBABILITY X 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.122 0.012 0.001 0.000 1 0.270 0.058 0.007 2 0.285 0.137 0.028 0.003 3 0.190 0.205 0.072 4 0.090 0.218 0.130 0.035 0.005 5 0.032 0.175 0.179 0.075 0.015 6 0.009 0.109 0.192 0.124 0.037 7 0.002 0.055 0.164 0.166 0.074 8 0.022 0.114 0.180 0.120 0.004 9 0.065 0.160 0.071 10 0.031 0.117 0.176 11 12 13 14 15 16 17 18 19 20 Binomial Table Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Using the Binomial Table Demonstration Problem 5.4 PROBABILITY X 0.1 0.2 0.3 0.4 0.122 0.012 0.001 0.000 1 0.270 0.058 0.007 2 0.285 0.137 0.028 0.003 3 0.190 0.205 0.072 4 0.090 0.218 0.130 0.035 5 0.032 0.175 0.179 0.075 6 0.009 0.109 0.192 0.124 7 0.002 0.055 0.164 0.166 8 0.022 0.114 0.180 9 0.065 0.160 10 0.031 0.117 11 0.071 12 0.004 13 0.015 14 0.005 15 16 17 18 19 20 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Binomial Distribution using Table: Demonstration Problem 5.3 PROBABILITY X 0.05 0.06 0.07 0.3585 0.2901 0.2342 1 0.3774 0.3703 0.3526 2 0.1887 0.2246 0.2521 3 0.0596 0.0860 0.1139 4 0.0133 0.0233 0.0364 5 0.0022 0.0048 0.0088 6 0.0003 0.0008 0.0017 7 0.0000 0.0001 0.0002 8 … 20 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

Excel’s Binomial Function 20 p = 0.06 X P(X) =BINOMDIST(A5,B$1,B$2,FALSE) 1 =BINOMDIST(A6,B$1,B$2,FALSE) 2 =BINOMDIST(A7,B$1,B$2,FALSE) 3 =BINOMDIST(A8,B$1,B$2,FALSE) 4 =BINOMDIST(A9,B$1,B$2,FALSE) 5 =BINOMDIST(A10,B$1,B$2,FALSE) 6 =BINOMDIST(A11,B$1,B$2,FALSE) 7 =BINOMDIST(A12,B$1,B$2,FALSE) 8 =BINOMDIST(A13,B$1,B$2,FALSE) 9 =BINOMDIST(A14,B$1,B$2,FALSE) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Graphs of Selected Binomial Distributions PROBABILITY X 0.1 0.5 0.9 0.656 0.063 0.000 1 0.292 0.250 0.004 2 0.049 0.375 3 4 P = 0.5 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 X P(X) P = 0.1 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 X P(X) P = 0.9 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 X P(X) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Poisson Distribution Describes discrete occurrences over a continuum or interval A discrete distribution Describes rare events Each occurrence is independent any other occurrences. The number of occurrences in each interval can vary from zero to infinity. The expected number of occurrences must hold constant throughout the experiment. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Poisson Distribution: Applications Arrivals at queuing systems airports -- people, airplanes, automobiles, baggage banks -- people, automobiles, loan applications computer file servers -- read and write operations Defects in manufactured goods number of defects per 1,000 feet of extruded copper wire number of blemishes per square foot of painted surface number of errors per typed page Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 33

Poisson Distribution Mean value Standard deviation Variance Probability function Mean value Standard deviation Variance Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

Poisson Distribution: Demonstration Problem 5.7 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Poisson Distribution: Probability Table X 0.5 1.5 1.6 3.0 3.2 6.4 6.5 7.0 8.0 0.6065 0.2231 0.2019 0.0498 0.0408 0.0017 0.0015 0.0009 0.0003 1 0.3033 0.3347 0.3230 0.1494 0.1304 0.0106 0.0098 0.0064 0.0027 2 0.0758 0.2510 0.2584 0.2240 0.2087 0.0340 0.0318 0.0223 0.0107 3 0.0126 0.1255 0.1378 0.2226 0.0726 0.0688 0.0521 0.0286 4 0.0016 0.0471 0.0551 0.1680 0.1781 0.1162 0.1118 0.0912 0.0573 5 0.0002 0.0141 0.0176 0.1008 0.1140 0.1487 0.1454 0.1277 0.0916 6 0.0000 0.0035 0.0047 0.0504 0.0608 0.1586 0.1575 0.1490 0.1221 7 0.0008 0.0011 0.0216 0.0278 0.1450 0.1462 0.1396 8 0.0001 0.0081 0.0111 0.1160 0.1188 9 0.0040 0.0825 0.0858 0.1014 0.1241 10 0.0013 0.0528 0.0558 0.0710 0.0993 11 0.0004 0.0307 0.0330 0.0452 0.0722 12 0.0164 0.0179 0.0263 0.0481 13 0.0089 0.0142 0.0296 14 0.0037 0.0041 0.0071 0.0169 15 0.0018 0.0033 0.0090 16 0.0006 0.0007 0.0014 0.0045 17 0.0021 18  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Poisson Distribution: Using the Poisson Tables X 0.5 1.5 1.6 3.0 0.6065 0.2231 0.2019 0.0498 1 0.3033 0.3347 0.3230 0.1494 2 0.0758 0.2510 0.2584 0.2240 3 0.0126 0.1255 0.1378 4 0.0016 0.0471 0.0551 0.1680 5 0.0002 0.0141 0.0176 0.1008 6 0.0000 0.0035 0.0047 0.0504 7 0.0008 0.0011 0.0216 8 0.0001 0.0081 9 0.0027 10 11 12  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Poisson Distribution: Using the Poisson Tables  X 0.5 1.5 1.6 3.0 0.6065 0.2231 0.2019 0.0498 1 0.3033 0.3347 0.3230 0.1494 2 0.0758 0.2510 0.2584 0.2240 3 0.0126 0.1255 0.1378 4 0.0016 0.0471 0.0551 0.1680 5 0.0002 0.0141 0.0176 0.1008 6 0.0000 0.0035 0.0047 0.0504 7 0.0008 0.0011 0.0216 8 0.0001 0.0081 9 0.0027 10 11 12 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Poisson Distribution: Using the Poisson Tables  X 0.5 1.5 1.6 3.0 0.6065 0.2231 0.2019 0.0498 1 0.3033 0.3347 0.3230 0.1494 2 0.0758 0.2510 0.2584 0.2240 3 0.0126 0.1255 0.1378 4 0.0016 0.0471 0.0551 0.1680 5 0.0002 0.0141 0.0176 0.1008 6 0.0000 0.0035 0.0047 0.0504 7 0.0008 0.0011 0.0216 8 0.0001 0.0081 9 0.0027 10 11 12 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Poisson Distribution: Graphs 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 1 2 3 4 5 6 7 8 0.02 0.04 0.06 0.08 0.12 0.14 0.16 10 12 14 16 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Excel’s Poisson Function 1.6 X P(X) =POISSON(D5,E$1,FALSE) 1 =POISSON(D6,E$1,FALSE) 2 =POISSON(D7,E$1,FALSE) 3 =POISSON(D8,E$1,FALSE) 4 =POISSON(D9,E$1,FALSE) 5 =POISSON(D10,E$1,FALSE) 6 =POISSON(D11,E$1,FALSE) 7 =POISSON(D12,E$1,FALSE) 8 =POISSON(D13,E$1,FALSE) 9 =POISSON(D14,E$1,FALSE) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Poisson Approximation of the Binomial Distribution Binomial probabilities are difficult to calculate when n is large. Under certain conditions binomial probabilities may be approximated by Poisson probabilities. Poisson approximation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 41

Poisson Approximation of the Binomial Distribution Error 0.2231 0.2181 -0.0051 1 0.3347 0.3372 0.0025 2 0.2510 0.2555 0.0045 3 0.1255 0.1264 0.0009 4 0.0471 0.0459 -0.0011 5 0.0141 0.0131 -0.0010 6 0.0035 0.0030 -0.0005 7 0.0008 0.0006 -0.0002 8 0.0001 0.0000 9 0.0498 0.1494 0.1493 0.2240 0.2241 0.1680 0.1681 0.1008 0.0504 0.0216 0.0081 0.0027 10 11 0.0002 12 13 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 42

Hypergeometric Distribution Sampling without replacement from a finite population The number of objects in the population is denoted N. Each trial has exactly two possible outcomes, success and failure. Trials are not independent X is the number of successes in the n trials The binomial is an acceptable approximation, if n < 5% N. Otherwise it is not. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Hypergeometric Distribution Probability function N is population size n is sample size A is number of successes in population x is number of successes in sample Mean value Variance and standard deviation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Hypergeometric Distribution: Probability Computations X = 8 n = 5 x 0.1028 1 0.3426 2 0.3689 3 0.1581 4 0.0264 5 0.0013 P(x) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Hypergeometric Distribution: Graph X = 8 n = 5 x 0.1028 1 0.3426 2 0.3689 3 0.1581 4 0.0264 5 0.0013 P(x) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 1 2 3 4 5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Hypergeometric Distribution: Demonstration Problem 5.11 X P(X) 0.0245 1 0.2206 2 0.4853 3 0.2696 N = 18 n = 3 A = 12 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Hypergeometric Distribution: Binomial Approximation (large n) Error 0.1028 0.1317 -0.0289 1 0.3426 0.3292 0.0133 2 0.3689 0.0397 3 0.1581 0.1646 -0.0065 4 0.0264 0.0412 -0.0148 5 0.0013 0.0041 -0.0028 P(x) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

Hypergeometric Distribution: Binomial Approximation (small n) P(x) Error 0.1289 0.1317 -0.0028 1 0.3306 0.3292 0.0014 2 0.3327 0.0035 3 0.1642 0.1646 -0.0004 4 0.0398 0.0412 -0.0014 5 0.0038 0.0041 -0.0003 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Excel’s Hypergeometric Function 24 A = 8 n = 5 X P(X) =HYPGEOMDIST(A6,B$3,B$2,B$1) 1 =HYPGEOMDIST(A7,B$3,B$2,B$1) 2 =HYPGEOMDIST(A8,B$3,B$2,B$1) 3 =HYPGEOMDIST(A9,B$3,B$2,B$1) 4 =HYPGEOMDIST(A10,B$3,B$2,B$1) =HYPGEOMDIST(A11,B$3,B$2,B$1) =SUM(B6:B11) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Business Statistics, 4e by Ken Black Chapter 6 Continuous Distributions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Understand concepts of the uniform distribution. Appreciate the importance of the normal distribution. Recognize normal distribution problems, and know how to solve them. Decide when to use the normal distribution to approximate binomial distribution problems, and know how to work them. Decide when to use the exponential distribution to solve problems in business, and know how to work them. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Uniform Distribution a b Area = 1 3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 3

Uniform Distribution of Lot Weights Area = 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Uniform Distribution Probability Area = 0.5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Uniform Distribution Mean and Standard Deviation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Characteristics of the Normal Distribution Continuous distribution Symmetrical distribution Asymptotic to the horizontal axis Unimodal A family of curves Area under the curve sums to 1. Area to right of mean is 1/2. Area to left of mean is 1/2. 1/2 X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Probability Density Function of the Normal Distribution X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Normal Curves for Different Means and Standard Deviations 20 30 40 50 60 70 80 90 100 110 120 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Standardized Normal Distribution A normal distribution with a mean of zero, and a standard deviation of one Z Formula standardizes any normal distribution Z Score computed by the Z Formula the number of standard deviations which a value is away from the mean s = 1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Z Table Second Decimal Place in Z 0.00 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.10 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.20 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.30 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.90 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 1.00 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.10 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.20 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 2.00 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 3.00 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 3.40 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998 3.50 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Table Lookup of a Standard Normal Probability -3 -2 -1 1 2 3 Z 0.00 0.01 0.02 0.00 0.0000 0.0040 0.0080 0.10 0.0398 0.0438 0.0478 0.20 0.0793 0.0832 0.0871 1.00 0.3413 0.3438 0.3461 1.10 0.3643 0.3665 0.3686 1.20 0.3849 0.3869 0.3888 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Applying the Z Formula Z 0.00 0.01 0.02 0.00 0.0000 0.0040 0.0080 0.00 0.0000 0.0040 0.0080 0.10 0.0398 0.0438 0.0478 1.00 0.3413 0.3438 0.3461 1.10 0.3643 0.3665 0.3686 1.20 0.3849 0.3869 0.3888 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Normal Approximation of the Binomial Distribution The normal distribution can be used to approximate binomial probabilities Procedure Convert binomial parameters to normal parameters Does the interval lie between 0 and n? If so, continue; otherwise, do not use the normal approximation. Correct for continuity Solve the normal distribution problem m s ± 3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Normal Approximation of Binomial: Parameter Conversion Conversion equations Conversion example: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Normal Approximation of Binomial: Interval Check 10 20 30 40 50 60 n 70 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Normal Approximation of Binomial: Correcting for Continuity Values Being Determined Correction X X X X X X +.50 -.50 +.05 -.50 and +.50 +.50 and -.50 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Normal Approximation of Binomial: Graphs 0.02 0.04 0.06 0.08 0.10 0.12 6 8 10 12 14 16 18 20 22 24 26 28 30 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Normal Approximation of Binomial: Computations 25 26 27 28 29 30 31 32 33 Total 0.0167 0.0096 0.0052 0.0026 0.0012 0.0005 0.0002 0.0001 0.0000 0.0361 X P(X) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

Exponential Distribution Continuous Family of distributions Skewed to the right X varies from 0 to infinity Apex is always at X = 0 Steadily decreases as X gets larger Probability function Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Graphs of Selected Exponential Distributions 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1 2 3 4 5 6 7 8     Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Exponential Distribution: Probability Computation 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1 2 3 4 5  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 33

Business Statistics, 4e by Ken Black Chapter 7 Sampling & Sampling Distributions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives x  p Determine when to use sampling instead of a census. Distinguish between random and nonrandom sampling. Decide when and how to use various sampling techniques. Be aware of the different types of error that can occur in a study. Understand the impact of the Central Limit Theorem on statistical analysis. Use the sampling distributions of and . x  p Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Reasons for Sampling Sampling can save money. Sampling can save time. For given resources, sampling can broaden the scope of the data set. Because the research process is sometimes destructive, the sample can save product. If accessing the population is impossible; sampling is the only option. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 3

Reasons for Taking a Census Eliminate the possibility that a random sample is not representative of the population. The person authorizing the study is uncomfortable with sample information. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Population Frame A list, map, directory, or other source used to represent the population Overregistration -- the frame contains all members of the target population and some additional elements Example: using the chamber of commerce membership directory as the frame for a target population of member businesses owned by women. Underregistration -- the frame does not contain all members of the target population. Example: using the chamber of commerce membership directory as the frame for a target population of all businesses. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Random Versus Nonrandom Sampling Every unit of the population has the same probability of being included in the sample. A chance mechanism is used in the selection process. Eliminates bias in the selection process Also known as probability sampling Nonrandom Sampling Every unit of the population does not have the same probability of being included in the sample. Open the selection bias Not appropriate data collection methods for most statistical methods Also known as nonprobability sampling Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Random Sampling Techniques Simple Random Sample Stratified Random Sample Proportionate Disportionate Systematic Random Sample Cluster (or Area) Sampling Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Simple Random Sample Number each frame unit from 1 to N. Use a random number table or a random number generator to select n distinct numbers between 1 and N, inclusively. Easier to perform for small populations Cumbersome for large populations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Simple Random Sample: Numbered Population Frame 01 Alaska Airlines 02 Alcoa 03 Ashland 04 Bank of America 05 BellSouth 06 Chevron 07 Citigroup 08 Clorox 09 Delta Air Lines 10 Disney 11 DuPont 12 Exxon Mobil 13 General Dynamics 14 General Electric 15 General Mills 16 Halliburton 17 IBM 18 Kellog 19 KMart 20 Lowe’s 21 Lucent 22 Mattel 23 Mead 24 Microsoft 25 Occidental Petroleum 26 JCPenney 27 Procter & Gamble 28 Ryder 29 Sears 30 Time Warner Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Simple Random Sampling: Random Number Table 9 4 3 7 8 6 1 5 2 N = 30 n = 6 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Simple Random Sample: Sample Members 01 Alaska Airlines 02 Alcoa 03 Ashland 04 Bank of America 05 BellSouth 06 Chevron 07 Citigroup 08 Clorox 09 Delta Air Lines 10 Disney 11 DuPont 12 Exxon Mobil 13 General Dynamics 14 General Electric 15 General Mills 16 Halliburton 17 IBM 18 Kellog 19 KMart 20 Lowe’s 21 Lucent 22 Mattel 23 Mead 24 Microsoft 25 Occidental Petroleum 26 JCPenney 27 Procter & Gamble 28 Ryder 29 Sears 30 Time Warner N = 30 n = 6 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Stratified Random Sample Population is divided into nonoverlapping subpopulations called strata A random sample is selected from each stratum Potential for reducing sampling error Proportionate -- the percentage of thee sample taken from each stratum is proportionate to the percentage that each stratum is within the population Disproportionate -- proportions of the strata within the sample are different than the proportions of the strata within the population Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Stratified Random Sample: Population of FM Radio Listeners 20 - 30 years old (homogeneous within) (alike) 30 - 40 years old 40 - 50 years old Hetergeneous (different) between Stratified by Age Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Systematic Sampling Convenient and relatively easy to administer Population elements are an ordered sequence (at least, conceptually). The first sample element is selected randomly from the first k population elements. Thereafter, sample elements are selected at a constant interval, k, from the ordered sequence frame. k = N n , where : sample size population size size of selection interval Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Systematic Sampling: Example Purchase orders for the previous fiscal year are serialized 1 to 10,000 (N = 10,000). A sample of fifty (n = 50) purchases orders is needed for an audit. k = 10,000/50 = 200 First sample element randomly selected from the first 200 purchase orders. Assume the 45th purchase order was selected. Subsequent sample elements: 245, 445, 645, . . . Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Cluster Sampling Population is divided into nonoverlapping clusters or areas Each cluster is a miniature, or microcosm, of the population. A subset of the clusters is selected randomly for the sample. If the number of elements in the subset of clusters is larger than the desired value of n, these clusters may be subdivided to form a new set of clusters and subjected to a random selection process. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Cluster Sampling Advantages More convenient for geographically dispersed populations Reduced travel costs to contact sample elements Simplified administration of the survey Unavailability of sampling frame prohibits using other random sampling methods Disadvantages Statistically less efficient when the cluster elements are similar Costs and problems of statistical analysis are greater than for simple random sampling Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Cluster Sampling Grand Forks Portland Fargo Buffalo Boise Pittsfield San Jose Boise Phoenix Denver Cedar Rapids Buffalo Louisville Atlanta Portland Milwaukee Kansas City San Diego Tucson Grand Forks Fargo Sherman- Dension Odessa- Midland Cincinnati Pittsfield Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Nonrandom Sampling Convenience Sampling: sample elements are selected for the convenience of the researcher Judgment Sampling: sample elements are selected by the judgment of the researcher Quota Sampling: sample elements are selected until the quota controls are satisfied Snowball Sampling: survey subjects are selected based on referral from other survey respondents Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Errors Data from nonrandom samples are not appropriate for analysis by inferential statistical methods. Sampling Error occurs when the sample is not representative of the population Nonsampling Errors Missing Data, Recording, Data Entry, and Analysis Errors Poorly conceived concepts , unclear definitions, and defective questionnaires Response errors occur when people so not know, will not say, or overstate in their answers Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Sampling Distribution of x Proper analysis and interpretation of a sample statistic requires knowledge of its distribution. Process of Inferential Statistics Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Distribution of a Small Finite Population Population Histogram 1 2 3 52.5 57.5 62.5 67.5 72.5 Frequency N = 8 54, 55, 59, 63, 68, 69, 70 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Sample Space for n = 2 with Replacement Mean 1 (54,54) 54.0 17 (59,54) 56.5 33 (64,54) 59.0 49 (69,54) 61.5 2 (54,55) 54.5 18 (59,55) 57.0 34 (64,55) 59.5 50 (69,55) 62.0 3 (54,59) 19 (59,59) 35 (64,59) 51 (69,59) 64.0 4 (54,63) 58.5 20 (59,63) 61.0 36 (64,63) 63.5 52 (69,63) 66.0 5 (54,64) 21 (59,64) 37 (64,64) 53 (69,64) 66.5 6 (54,68) 22 (59,68) 38 (64,68) 54 (69,68) 68.5 7 (54,69) 23 (59,69) 39 (64,69) 55 (69,69) 69.0 8 (54,70) 24 (59,70) 64.5 40 (64,70) 67.0 56 (69,70) 69.5 9 (55,54) 25 (63,54) 41 (68,54) 57 (70,54) 10 (55,55) 55.0 26 (63,55) 42 (68,55) 58 (70,55) 62.5 11 (55,59) 27 (63,59) 43 (68,59) 59 (70,59) 12 (55,63) 28 (63,63) 63.0 44 (68,63) 65.5 60 (70,63) 13 (55,64) 29 (63,64) 45 (68,64) 61 (70,64) 14 (55,68) 30 (63,68) 46 (68,68) 68.0 62 (70,68) 15 (55,69) 31 (63,69) 47 (68,69) 63 (70,69) 16 (55,70) 32 (63,70) 48 (68,70) 64 (70,70) 70.0 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Distribution of the Sample Means Sampling Distribution Histogram 5 10 15 20 53.75 56.25 58.75 61.25 63.75 66.25 68.75 71.25 Frequency Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

1,800 Randomly Selected Values from an Exponential Distribution 50 100 150 200 250 300 350 400 450 .5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 X F r e q u n c y Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Means of 60 Samples (n = 2) from an Exponential Distribution q u n c y 1 2 3 4 5 6 7 8 9 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 x Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Means of 60 Samples (n = 5) from an Exponential Distribution q u n c y x 1 2 3 4 5 6 7 8 9 10 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Means of 60 Samples (n = 30) from an Exponential Distribution 2 4 6 8 10 12 14 16 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 F r e q u n c y x Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

1,800 Randomly Selected Values from a Uniform Distribution X F r e q u n c y 50 100 150 200 250 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Means of 60 Samples (n = 2) from a Uniform Distribution q u n c y x 1 2 3 4 5 6 7 8 9 10 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

Means of 60 Samples (n = 5) from a Uniform Distribution q u n c y x 2 4 6 8 10 12 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Means of 60 Samples (n = 30) from a Uniform Distribution q u n c y x 5 10 15 20 25 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Central Limit Theorem x s n For sufficiently large sample sizes (n  30), the distribution of sample means , is approximately normal; the mean of this distribution is equal to , the population mean; and its standard deviation is , regardless of the shape of the population distribution. x s n Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 33

Central Limit Theorem Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

Distribution of Sample Means for Various Sample Sizes Exponential Population n = 2 n = 5 n = 30 Uniform Population n = 2 n = 5 n = 30 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 36

Distribution of Sample Means for Various Sample Sizes Normal Population n = 2 n = 5 n = 30 U Shaped Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 37

Sampling from a Normal Population The distribution of sample means is normal for any sample size. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Z Formula for Sample Means Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 38

Solution to Tire Store Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Graphic Solution to Tire Store Example 87 85 .5000 .4207 Z 1.41 .5000 .4207 Equal Areas of .0793 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Graphic Solution for Demonstration Problem 7.1 448 X 441 446 .2486 .4901 .2415 Z -2.33 -.67 .2486 .4901 .2415 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 41

Sampling from a Finite Population without Replacement In this case, the standard deviation of the distribution of sample means is smaller than when sampling from an infinite population (or from a finite population with replacement). The correct value of this standard deviation is computed by applying a finite correction factor to the standard deviation for sampling from a infinite population. If the sample size is less than 5% of the population size, the adjustment is unnecessary. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 42

Sampling from a Finite Population Finite Correction Factor Modified Z Formula Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 43

Finite Correction Factor for Selected Sample Sizes Population Sample Sample % Value of Size (N) Size (n) of Population Correction Factor 6,000 30 0.50% 0.998 6,000 100 1.67% 0.992 6,000 500 8.33% 0.958 2,000 30 1.50% 0.993 2,000 100 5.00% 0.975 2,000 500 25.00% 0.866 500 30 6.00% 0.971 500 50 10.00% 0.950 500 100 20.00% 0.895 200 30 15.00% 0.924 200 50 25.00% 0.868 200 75 37.50% 0.793 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 44

Sampling Distribution of  Sample Proportion Sampling Distribution Approximately normal if nP > 5 and nQ > 5 (P is the population proportion and Q = 1 - P.) The mean of the distribution is P. The standard deviation of the distribution is P Q n  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 47

Z Formula for Sample Proportions Q n where     :  sample proportion sample size population proportion 1 5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 48

Solution for Demonstration Problem 7.3 Population Parameters = . - Sample P Q n X p Z 10 1 90 80 12 15     ( )       P Z ( . ) 1 49 5 4319 0681  Q n 15 (. 10 90 80 05 0335 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 49

Graphic Solution for Demonstration Problem 7.3 0.15 0.10 .5000 .4319 ^ Z 1.49 .5000 .4319 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 50

Business Statistics, 4e by Ken Black Chapter 8 Statistical Inference: Estimation for Single Populations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Know the difference between point and interval estimation. Estimate a population mean from a sample mean when s is known. Estimate a population mean from a sample mean when s is unknown. Estimate a population proportion from a sample proportion. Estimate the population variance from a sample variance. Estimate the minimum sample size necessary to achieve given statistical goals. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Statistical Estimation Point estimate -- the single value of a statistic calculated from a sample Interval Estimate -- a range of values calculated from a sample statistic(s) and standardized statistics, such as the z. Selection of the standardized statistic is determined by the sampling distribution. Selection of critical values of the standardized statistic is determined by the desired level of confidence. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 3

Confidence Interval to Estimate  when  is Known Point estimate Interval Estimate Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Distribution of Sample Means for (1-)% Confidence  X  Z Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Distribution of Sample Means for (1-)% Confidence  X Z Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Distribution of Sample Means for (1-)% Confidence  X Z Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Distribution of Sample Means for 95% Confidence  .4750 X 95% .025 Z 1.96 -1.96 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

95% Confidence Interval for  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

95% Confidence Intervals for  X 95% Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

95% Confidence Intervals for  Is our interval, 143.22  162.78, in the red? 95%  X X X X X X X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Demonstration Problem 8.1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Demonstration Problem 8.2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Confidence Interval to Estimate  when n is Large and  is Known Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Car Rental Firm Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Z Values for Some of the More Common Levels of Confidence 90% 95% 98% 99% Confidence Level z Value 1.645 1.96 2.33 2.575 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Estimating the Mean of a Normal Population: Unknown  The population has a normal distribution. The value of the population standard deviation is unknown. z distribution is not appropriate for these conditions t distribution is appropriate Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

The t Distribution Developed by British statistician, William Gosset A family of distributions -- a unique distribution for each value of its parameter, degrees of freedom (d.f.) Symmetric, Unimodal, Mean = 0, Flatter than a z t formula Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Comparison of Selected t Distributions to the Standard Normal -3 -2 -1 1 2 3 Standard Normal t (d.f. = 25) t (d.f. = 1) t (d.f. = 5) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Table of Critical Values of t df t0.100 t0.050 t0.025 t0.010 t0.005 1 3.078 6.314 12.706 31.821 63.656 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 40 1.303 1.684 2.021 2.423 2.704 60 1.296 1.671 2.000 2.390 2.660 120 1.289 1.658 1.980 2.358 2.617 1.282 1.645 1.960 2.327 2.576  t   With df = 24 and a = 0.05, ta = 1.711. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Confidence Intervals for  of a Normal Population: Unknown  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

Solution for Demonstration Problem 8.3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

Solution for Demonstration Problem 8.3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

Comp Time: Excel Normal View Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Comp Time: Excel Formula View B C D E F 1 Comp Time Data 2 6 21 17 20 7 3 8 16 29 12 4 11 9 25 15 5 n = =COUNT(A2:F4) Mean = =AVERAGE(A2:F4) S = =STDEV(A2:F4) Std Error = =B8/SQRT(B6) 10 a = 0.1 df = =B6-1 13 t = =TINV(B11,B12) 14 =B7-B13*B9 £ m £ =B7+B13*B9 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Confidence Interval to Estimate the Population Proportion Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Solution for Demonstration Problem 8.5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Population Variance Variance is an inverse measure of the group’s homogeneity. Variance is an important indicator of total quality in standardized products and services. Managers improve processes to reduce variance. Variance is a measure of financial risk. Variance of rates of return help managers assess financial and capital investment alternatives. Variability is a reality in global markets. Productivity, wages, and costs of living vary between regions and nations. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Estimating the Population Variance Population Parameter  Estimator of   formula for Single Variance Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Confidence Interval for 2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Selected 2 Distributions df = 3 df = 5 df = 10 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

2 Table df = 5 0.10 9.23635 With df = 5 and a = 0.10, c2 = 9.23635 df 0.975 0.950 0.100 0.050 0.025 1 9.82068E-04 3.93219E-03 2.70554 3.84146 5.02390 2 0.0506357 0.102586 4.60518 5.99148 7.37778 3 0.2157949 0.351846 6.25139 7.81472 9.34840 4 0.484419 0.710724 7.77943 9.48773 11.14326 5 0.831209 1.145477 9.23635 11.07048 12.83249 6 1.237342 1.63538 10.6446 12.5916 14.4494 7 1.689864 2.16735 12.0170 14.0671 16.0128 8 2.179725 2.73263 13.3616 15.5073 17.5345 9 2.700389 3.32512 14.6837 16.9190 19.0228 10 3.24696 3.94030 15.9872 18.3070 20.4832 20 9.59077 10.8508 28.4120 31.4104 34.1696 21 10.28291 11.5913 29.6151 32.6706 35.4789 22 10.9823 12.3380 30.8133 33.9245 36.7807 23 11.6885 13.0905 32.0069 35.1725 38.0756 24 12.4011 13.8484 33.1962 36.4150 39.3641 25 13.1197 14.6114 34.3816 37.6525 40.6465 70 48.7575 51.7393 85.5270 90.5313 95.0231 80 57.1532 60.3915 96.5782 101.8795 106.6285 90 65.6466 69.1260 107.5650 113.1452 118.1359 100 74.2219 77.9294 118.4980 124.3421 129.5613 With df = 5 and a = 0.10, c2 = 9.23635 5 10 15 20 0.10 df = 5 9.23635 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Two Table Values of 2 df = 7 .95 .05 2.16735 14.0671 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 18 20 df = 7 .05 .95 2.16735 14.0671 df 0.950 0.050 1 3.93219E-03 3.84146 2 0.102586 5.99148 3 0.351846 7.81472 4 0.710724 9.48773 5 1.145477 11.07048 6 1.63538 12.5916 7 2.16735 14.0671 8 2.73263 15.5073 9 3.32512 16.9190 10 3.94030 18.3070 20 10.8508 31.4104 21 11.5913 32.6706 22 12.3380 33.9245 23 13.0905 35.1725 24 13.8484 36.4150 25 14.6114 37.6525 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

90% Confidence Interval for 2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 33

Solution for Demonstration Problem 8.6 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

Determining Sample Size when Estimating  z formula Error of Estimation (tolerable error) Estimated Sample Size Estimated  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Sample Size When Estimating : Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 36

Solution for Demonstration Problem 8.7 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 37

Determining Sample Size when Estimating p z formula Error of Estimation (tolerable error) Estimated Sample Size Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 38

Solution for Demonstration Problem 8.8 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Determining Sample Size when Estimating p with No Prior Information 0.5 0.4 0.3 0.2 0.1 pq 0.25 0.24 0.21 0.16 0.09 P n 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z = 1.96 E = 0.05 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Example: Determining n when Estimating p with No Prior Information Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 41

Business Statistics, 4e by Ken Black Chapter 9 Statistical Inference: Hypothesis Testing for Single Populations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Understand the logic of hypothesis testing, and know how to establish null and alternate hypotheses. Understand Type I and Type II errors, and know how to solve for Type II errors. Know how to implement the HTAB system to test hypotheses. Test hypotheses about a single population mean when s is known. Test hypotheses about a single population mean when s is unknown. Test hypotheses about a single population proportion. Test hypotheses about a single population variance. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Types of Hypotheses Research Hypothesis Statistical Hypotheses a statement of what the researcher believes will be the outcome of an experiment or a study. Statistical Hypotheses a more formal structure derived from the research hypothesis. Substantive Hypotheses a statistically significant difference does not imply or mean a material, substantive difference. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Example Research Hypotheses Older workers are more loyal to a company Companies with more than $1 billion of assets spend a higher percentage of their annual budget on advertising than do companies with less than $1 billion of assets. The price of scrap metal is a good indicator of the industrial production index six months later. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Statistical Hypotheses Two Parts a null hypothesis an alternative hypothesis Null Hypothesis – nothing new is happening Alternative Hypothesis – something new is happening Notation null: H0 alternative: Ha Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Null and Alternative Hypotheses The Null and Alternative Hypotheses are mutually exclusive. Only one of them can be true. The Null and Alternative Hypotheses are collectively exhaustive. They are stated to include all possibilities. (An abbreviated form of the null hypothesis is often used.) The Null Hypothesis is assumed to be true. The burden of proof falls on the Alternative Hypothesis. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Null and Alternative Hypotheses: Example A manufacturer is filling 40 oz. packages with flour. The company wants the package contents to average 40 ounces. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

One-tailed and Two-tailed Tests One-tailed Tests Two-tailed Test Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

HTAB System to Test Hypotheses Task 1: HYPOTHESIZE Task 2: TEST Task 3: TAKE STATISTICAL ACTION Task 4: DETERMINING THE BUSINESS IMPLICATIONS Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Steps in Testing Hypotheses 1. Establish hypotheses: state the null and alternative hypotheses. 2. Determine the appropriate statistical test and sampling distribution. 3. Specify the Type I error rate ( 4. State the decision rule. 5. Gather sample data. 6. Calculate the value of the test statistic. 7. State the statistical conclusion. 8. Make a managerial decision. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

HTAB Paradigm – Task 1 Task 1: Hypotheses Step 1. Establish hypotheses: state the null and alternative hypotheses. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

HTAB Paradigm – Task 2 Task 2: Test Step 2. Determine the appropriate statistical test and sampling distribution. Step 3. Specify the Type I error rate ( Step 4. State the decision rule. Step 5. Gather sample data. Step 6. Calculate the value of the test statistic. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

HTAB Paradigm – Task 3 Task 3: Take Statistical Action Step 7. State the statistical conclusion. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

HTAB Paradigm – Task 4 Task 4: Determine the business implications Step 8. Make a managerial decision. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Rejection and Non Rejection Regions =40 oz Non Rejection Region Rejection Region Critical Value Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Type I and Type II Errors Type I Error Rejecting a true null hypothesis The probability of committing a Type I error is called , the level of significance. Type II Error Failing to reject a false null hypothesis The probability of committing a Type II error is called . Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Decision Table for Hypothesis Testing ( ) Null True Null False Fail to reject null Correct Decision Type II error  Reject null Type I error  Correct Decision Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

One-tailed Tests =40 oz Rejection Region Non Rejection Region Critical Value =40 oz Rejection Region Non Rejection Region Critical Value Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Two-tailed Tests Rejection Region Non Rejection Region =12 oz Critical Values Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

CPA Net Income Example: Two-tailed Test (Part 1) Rejection Region Non Rejection Region =0 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

CPA Net Income Example: Two-tailed Test (Part 2) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

CPA Net Income Example: Critical Value Method (Part 1) Rejection Region Non Rejection Region =0 72,223 77,605 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

CPA Net Income Example: Critical Value Method (Part 2) Rejection Region Non Rejection Region =0 72,223 77,605 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Demonstration Problem 9.1: z Test (Part 1) Rejection Region Non Rejection Region =.05 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Demonstration Problem 9.1: z Test (Part 2) Rejection Region Non Rejection Region =.05 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Demonstration Problem 9.1: Critical Value (Part 1) Rejection Region Non Rejection Region =.05 4.30 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Demonstration Problem 9.1: Critical Value (Part 2) Rejection Region Non Rejection Region =.05 4.30 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Demonstration Problem 9.1: Using the p-Value Rejection Region Non Rejection Region =.05 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Demonstration Problem 9.1: MINITAB Test of mu = 4.300 vs mu < 4.300 The assumed sigma = 0.574 Variable N MEAN STDEV SE MEAN Z P VALUE Ratings 32 4.156 0.574 0.101 -1.42 0.078 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Demonstration Problem 9.1: Excel (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Demonstration Problem 9.1: Excel (Part 2) H0: m = 4.3 Ha: m < 3 4 5 n = =COUNT(A4:H7) a = 0.05 Mean = =AVERAGE(A4:H7) S = =STDEV(A4:H7) Std Error = =B12/SQRT(B9) Z = =(B11-B1)/B13 p-Value =NORMSDIST(B14) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Two-tailed Test:  Unknown,  = .05 (Part 1) Weights in Pounds of a Sample of 20 Plates 22.6 22.2 23.2 27.4 24.5 27.0 26.6 28.1 26.9 24.9 26.2 25.3 23.1 24.2 26.1 25.8 30.4 28.6 23.5 23.6 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Two-tailed Test:  Unknown,  = .05 (part 2) Critical Values Non Rejection Region Rejection Regions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

Two-tailed Test:  Unknown,  = .05 (part 3) Critical Values Non Rejection Region Rejection Regions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

MINITAB Computer Printout for the Machine Plate Example Test of mu = 25.000 vs mu not = 25.000 Variable N MEAN STDEV SE MEAN T P VALUE Platewt 20 25.510 2.193 0.490 1.04 0.31 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

Machine Plate Example: Excel (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Machine Plate Example: Excel (Part 2) B C D E 1 H0: m = 25 2 Ha: m ¹ 3 4 22.6 22.2 23.2 27.4 24.5 5 27 26.6 28.1 26.9 24.9 6 26.2 25.3 23.1 24.2 26.1 7 25.8 30.4 28.6 23.5 23.6 8 9 n = =COUNT(A4:E7) 10 a = 0.05 11 Mean = =AVERAGE(A4:E7) 12 S = =STDEV(A4:E7) 13 Std Error = =B12/SQRT(B9) 14 t = =(B11-B1)/B13 15 p-Value =TDIST(B14,B9-1,2) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Demonstration Problem 9.2 (Part 1) Size in Acres of 23 Farms 445 489 474 505 553 477 545 463 466 557 502 449 438 500 466 477 557 433 545 511 590 561 560 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Demonstration Problem 9.2 (Part 2) Critical Value Non Rejection Region Rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Demonstration Problem 9.2 (Part 3) Critical Value Non Rejection Region Rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

z Test of Population Proportion Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Testing Hypotheses about a Proportion: Manufacturer Example (Part 1) Critical Values Non Rejection Region Rejection Regions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Testing Hypotheses about a Proportion: Manufacturer Example (Part 2) Critical Values Non Rejection Region Rejection Regions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

Demonstration Problem 9.3 (Part 1) Critical Value Non Rejection Region Rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Demonstration Problem 9.3 (Part 2) Critical Value Non Rejection Region Rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Hypothesis Test for 2: Demonstration Problem 9.4 (Part 1) df = 15 .05 .95 7.26094 24.9958 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 33

Hypothesis Test for 2: Demonstration Problem 9.4 (Part 2) df = 15 .05 .95 7.26094 24.9958 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

Solving for Type II Errors: The Beverage Example Rejection Region Non Rejection Region =0 =.05 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Type II Error for Beverage Example with  =11.99 oz =.05 Reject Ho Do Not Reject Ho  Ho is True Ho is False 95% =.8023 Correct Decision Type I Error Type II 19.77%      Z0 Z1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 36

Type II Error for Demonstration Problem 9.5, with =11.96 oz =.05  Ho is True Ho is False 95% Reject Ho Do Not Reject Ho =.0708 Correct Decision Type I Error Type II 92.92%     Z0 Z1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 37

 Values and Power Values for the Soft-Drink Example 11.999 .94 .06 11.995 .89 .11 11.990 .80 .20 11.980 .53 .47 11.970 .24 .76 11.960 .07 .93 11.950 .01 .99 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 38

Operating Characteristic Curve for the Soft-Drink Example 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11.95 11.96 11.97 11.98 11.99 12 Probability  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Power Curve for the Soft-Drink Example 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11.95 11.96 11.97 11.98 11.99 12 Probability  Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Business Statistics, 4e by Ken Black Chapter 10 Statistical Inferences about Two Populations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Test hypotheses and construct confidence intervals about the difference in two population means using the Z statistic. Test hypotheses and construct confidence intervals about the difference in two population means using the t statistic. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Learning Objectives Test hypotheses and construct confidence intervals about the difference in two related populations. Test hypotheses and construct confidence intervals about the differences in two population proportions. Test hypotheses and construct confidence intervals about two population variances. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Sampling Distribution of the Difference Between Two Sample Means Population 1 Population 2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 3

Sampling Distribution of the Difference between Two Sample Means Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Z Formula for the Difference in Two Sample Means When 12 and22 are known and Independent Samples Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Hypothesis Testing for Differences Between Means: The Wage Example (part 1) Rejection Region Non Rejection Region Critical Values H o a : 1 2 m - = ¹ Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Hypothesis Testing for Differences Between Means: The Wage Example (part 2) Rejection Region Non Rejection Region Critical Values Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Hypothesis Testing for Differences Between Means: The Wage Example (part 3) Advertising Managers 74.256 57.791 71.115 96.234 65.145 67.574 89.807 96.767 59.621 93.261 77.242 62.483 103.030 67.056 69.319 74.195 64.276 35.394 75.932 74.194 86.741 80.742 65.360 57.351 39.672 73.904 45.652 54.270 93.083 59.045 63.384 68.508 Auditing Managers 69.962 77.136 43.649 55.052 66.035 63.369 57.828 54.335 59.676 63.362 42.494 54.449 37.194 83.849 46.394 99.198 67.160 71.804 61.254 37.386 72.401 73.065 59.505 56.470 48.036 72.790 67.814 60.053 71.351 71.492 66.359 58.653 61.261 63.508 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Hypothesis Testing for Differences between Means: The Wage Example (part 4) Rejection Region Non Rejection Region Critical Values Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Difference Between Means: Using Excel z-Test: Two Sample for Means   Adv Mgr Auditing Mgr Mean 70.7001 62.187 Known Variance 264.164 166.411 Observations 32 34 Hypothesized Mean Difference z 2.35 P(Z<=z) one-tail 0.0094 z Critical one-tail 1.64 P(Z<=z) two-tail 0.0189 z Critical two-tail 1.960 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Demonstration Problem 10.1 (part 1) Non Rejection Region Critical Value Rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Demonstration Problem 10.1 (part 2) Non Rejection Region Critical Value Rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Confidence Interval to Estimate 1 - 2 When 1, 2 are known Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Demonstration Problem 10.2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

The t Test for Differences in Population Means Each of the two populations is normally distributed. The two samples are independent. The values of the population variances are unknown. The variances of the two populations are equal. 12 = 22 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

t Formula to Test the Difference in Means Assuming 12 = 22 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Hernandez Manufacturing Company (part 1) Rejection Region Non Rejection Region Critical Values Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Hernandez Manufacturing Company (part 2) Training Method A 56 51 45 47 52 43 42 53 50 48 44 Training Method B 59 52 53 54 57 56 55 64 65 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Hernandez Manufacturing Company (part 3) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

MINITAB Output for Hernandez New-Employee Training Problem Twosample T for method A vs method B N Mean StDev SE Mean method A 15 47.73 4.42 1.1 method B 12 56.60 4.27 1.2 95% C.I. for mu method A - mu method B: (-12.2, -5.3) T-Test mu method A = mu method B (vs not =): T = -5.20 P=0.0000 DF = 25 Both use Pooled StDev = 4.35 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

EXCEL Output for Hernandez New-Employee Training Problem t-Test: Two-Sample Assuming Equal Variances Variable 1 Variable 2 Mean 4 7.73 56.5 Variance 19.495 18.27 Observations 15 12 Pooled Variance 18.957 Hypothesized Mean Difference df 25 t Stat - 5.20 P(T<=t) one-tail 1.12E-05 t Critical one-tail 1.71 P(T<=t) two-tail 2.23E-05 t Critical two-tail 2.06 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Confidence Interval to Estimate 1 - 2 when 12 and 22 are unknown and 12 = 22 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Dependent Samples Before and after measurements on the same individual Studies of twins Studies of spouses Individual 1 2 3 4 5 6 7 Before 32 11 21 17 30 38 14 After 39 15 35 13 41 22 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Formulas for Dependent Samples Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

P/E Ratios for Nine Randomly Selected Companies Company 2001 P/E Ratio 2002 P/E Ratio 1 8.9 12.7 2 38.1 45.4 3 43.0 10.0 4 34.0 27.2 5 34.5 22.8 6 15.2 24.1 7 20.3 32.3 8 19.9 40.1 9 61.9 106.5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies Rejection Region Non Rejection Region Critical Value Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies Company 2001 P/E Ratio 2002 P/E Ratio d 1 8.9 12.7 -3.8 2 38.1 45.4 -7.3 3 43.0 10.0 33.0 4 34.0 27.2 6.8 5 34.5 22.8 11.7 6 15.2 24.1 -8.9 7 20.3 32.3 -12.0 8 19.9 40.1 -20.2 9 61.9 106.5 -44.6 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies t-Test: Paired Two Sample for Means   2001 P/E Ratio 2002 P/E Ratio Mean 30.64 35.68 Variance 268.1 837.5 Observations 9 Pearson Correlation 0.674 Hypothesized Mean Difference df 8 t Stat -0.7 P(T<=t) one-tail 0.252 t Critical one-tail 1.86 P(T<=t) two-tail 0.504 t Critical two-tail 2.306 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Hypothesis Testing with Dependent Samples: Demonstration Problem 10.5 Individual 1 2 3 4 5 6 7 Before 32 11 21 17 30 38 14 After 39 15 35 13 41 22 d -7 -4 -14 -11 -1 -8 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Hypothesis Testing with Dependent Samples: Demonstration Problem 10.5 Rejection Region Non Rejection Region Critical Value Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Hypothesis Testing with Dependent Samples: Demonstration Problem 10.5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 33

Confidence Intervals for Mean Difference for Related Samples Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Difference in Number of New-House Sales Realtor May 2001 May 2002 d 1 8 11 -3 2 19 30 -11 3 5 6 -1 4 9 13 -4 -2 7 15 17 -6 12 10 -7 14 22 -8 16 18 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Confidence Interval for Mean Difference in Number of New-House Sales Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Sampling Distribution of Differences in Sample Proportions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Z Formula for the Difference in Two Population Proportions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Z Formula to Test the Difference in Population Proportions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 41

Testing the Difference in Population Proportions (Demonstration Problem 10.6) Rejection Region Non Rejection Region Critical Values Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 42

Testing the Difference in Population Proportions (Demonstration Problem 10.6) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 43

Confidence Interval to Estimate p1 - p2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 44

Example Problem: When do men shop for groceries? Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 45

F Test for Two Population Variances Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 46

F Distribution with 1 = 10 and 2 = 8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

A Portion of the F Distribution Table for  = 0.025 Numerator Degrees of Freedom Denominator Degrees of Freedom 1 2 3 4 5 6 7 8 9 647.79 799.48 864.15 899.60 921.83 937.11 948.20 956.64 963.28 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 8.07 6.54 5.89 5.29 5.12 4.99 4.90 4.82 7.57 6.06 5.42 5.05 4.65 4.53 4.43 4.36 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 12 6.55 5.10 4.12 3.89 3.73 3.61 3.51 3.44 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 48

Sheet Metal Example: Hypothesis Test for Equality of Two Population Variances (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 49

Sheet metal Manufacturer (Part 2) Rejection Regions Critical Values Non Rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Sheet Metal Example (Part 3) Machine 1 22.3 21.8 22.2 21.9 21.6 22.4 22.5 Machine 2 22.0 22.1 21.7 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 51

Business Statistics, 4e by Ken Black Chapter 11 Analysis of Variance & Design of Experiments Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Understand the differences between various experimental designs and when to use them. Compute and interpret the results of a one-way ANOVA. Compute and interpret the results of a random block design. Compute and interpret the results of a two-way ANOVA. Understand and interpret interaction. Know when and how to use multiple comparison techniques. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Introduction to Design of Experiments, #1 Experimental Design - a plan and a structure to test hypotheses in which the researcher controls or manipulates one or more variables. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

Introduction to Design of Experiments, #2 Independent Variable Treatment variable is one that the experimenter controls or modifies in the experiment. Classification variable is a characteristic of the experimental subjects that was present prior to the experiment, and is not a result of the experimenter’s manipulations or control. Levels or Classifications are the subcategories of the independent variable used by the researcher in the experimental design. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Introduction to Design of Experiments, #3 Dependent Variable - the response to the different levels of the independent variables. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Three Types of Experimental Designs Completely Randomized Design Randomized Block Design Factorial Experiments Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Completely Randomized Design Machine Operator Valve Opening Measurements 1 . 2 4 3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Valve Openings by Operator 1 2 3 4 6.33 6.26 6.44 6.29 6.36 6.38 6.23 6.31 6.58 6.19 6.27 6.54 6.21 6.4 6.56 6.5 6.34 6.22 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Analysis of Variance: Assumptions Observations are drawn from normally distributed populations. Observations represent random samples from the populations. Variances of the populations are equal. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

One-Way ANOVA: Procedural Overview Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

One-Way ANOVA: Sums of Squares Definitions Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Partitioning Total Sum of Squares of Variation SST (Total Sum of Squares) SSC (Treatment Sum of Squares) SSE (Error Sum of Squares) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

One-Way ANOVA: Computational Formulas Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

One-Way ANOVA: Preliminary Calculations 1 2 3 4 6.33 6.26 6.44 6.29 6.36 6.38 6.23 6.31 6.58 6.19 6.27 6.54 6.21 6.4 6.56 6.5 6.34 6.22 Tj T1 = 31.59 T2 = 50.22 T3 = 45.42 T4 = 24.92 T = 152.15 nj n1 = 5 n2 = 8 n3 = 7 n4 = 4 N = 24 Mean 6.318000 6.277500 6.488571 6.230000 6.339583 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

One-Way ANOVA: Sum of Squares Calculations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

One-Way ANOVA: Sum of Squares Calculations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

One-Way ANOVA: Mean Square and F Calculations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Analysis of Variance for Valve Openings Source of Variance df SS MS F Between 3 0.23658 0.078860 10.18 Error 20 0.15492 0.007746 Total 23 0.39150 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

A Portion of the F Table for  = 0.05 df1 df 2 1 2 3 4 5 6 7 8 9 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 … 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.39 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.37 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

One-Way ANOVA: Procedural Summary Rejection Region  Critical Value Non rejection Region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Excel Output for the Valve Opening Example Anova: Single Factor SUMMARY Groups Count Sum Average Variance Operator 1 5 31.59 6.318 0.00277 Operator 2 8 50.22 6.2775 0.0110786 Operator 3 7 45.42 6.488571429 0.0101143 Operator 4 4 24.92 6.23 0.0018667 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 0.236580119 3 0.07886004 10.181025 0.00028 3.09839 Within Groups 0.154915714 20 0.007745786 Total 0.391495833 23   Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Multiple Comparison Tests An analysis of variance (ANOVA) test is an overall test of differences among groups. Multiple Comparison techniques are used to identify which pairs of means are significantly different given that the ANOVA test reveals overall significance. Tukey’s honestly significant difference (HSD) test requires equal sample sizes Tukey-Kramer Procedure is used when sample sizes are unequal. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

Tukey’s Honestly Significant Difference (HSD) Test Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

Data from Demonstration Problem 11.1 PLANT (Employee Age) 1 2 3 29 32 25 27 33 24 30 31 24 27 34 25 28 30 26 Group Means 28.2 32.0 24.8 nj 5 5 5 C = 3 dfE = N - C = 12 MSE = 1.63 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

q Values for  = .01 Degrees of Freedom 1 2 3 4 . 11 12 5 90 135 164 186 14 19 22.3 24.7 8.26 10.6 12.2 13.3 6.51 8.12 9.17 9.96 4.39 5.14 5.62 5.97 4.32 5.04 5.50 5.84 ... Number of Populations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Tukey’s HSD Test for the Employee Age Data Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Tukey-Kramer Procedure: The Case of Unequal Sample Sizes Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Freighter Example: Means and Sample Sizes for the Four Operators 1 5 6.3180 2 8 6.2775 3 7 6.4886 4 6.2300 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Tukey-Kramer Results for the Four Operators Pair Critical Difference |Actual Differences| 1 and 2 .1405 .0405 1 and 3 .1443 .1706* 1 and 4 .1653 .0880 2 and 3 .1275 .2111* 2 and 4 .1509 .0475 3 and 4 .1545 .2586* *denotes significant at  .05 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Partitioning the Total Sum of Squares in the Randomized Block Design SST (Total Sum of Squares) SSC (Treatment Sum of Squares) SSE (Error Sum of Squares) SSR (Sum of Squares Blocks) SSE’ Error) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

A Randomized Block Design Individual observations . Single Independent Variable Blocking Variable Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

Randomized Block Design Treatment Effects: Procedural Overview Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Randomized Block Design: Computational Formulas Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

Randomized Block Design: Tread-Wear Example Supplier 1 2 3 4 Slow Medium Fast Block Means ( ) 3.7 4.5 3.1 3.77 3.4 3.9 2.8 3.37 3.5 4.1 3.0 3.53 3.2 2.6 3.10 5 Treatment Means( ) 4.8 4.03 3.54 4.16 2.98 3.56 Speed n = 5 N = 15 C = 3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Randomized Block Design: Sum of Squares Calculations (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 36

Randomized Block Design: Sum of Squares Calculations (Part 2) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Randomized Block Design: Mean Square Calculations Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Analysis of Variance for the Tread-Wear Example Source of Variance SS df MS F Treatment 3.484 2 1.742 96.78 Block 1.549 4 0.387 21.50 Error 0.143 8 0.018 Total 5.176 14 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Randomized Block Design Treatment Effects: Procedural Summary Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Randomized Block Design Blocking Effects: Procedural Overview Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 41

Excel Output for Tread-Wear Example: Randomized Block Design Anova: Two-Factor Without Replication SUMMARY Count Sum Average Variance Suplier 1 3 11.3 3.7666667 0.4933333 Suplier 2 10.1 3.3666667 0.3033333 Suplier 3 10.6 3.5333333 Suplier 4 9.3 3.1 0.21 Suplier 5 12.1 4.0333333 0.5033333 Slow 5 17.7 3.54 0.073 Medium 20.8 4.16 0.258 Fast 14.9 2.98 0.092 ANOVA Source of Variation SS df MS F P-value F crit Rows 1.5493333 4 0.3873333 21.719626 0.0002357 7.0060651 Columns 3.484 2 1.742 97.682243 2.395E-06 8.6490672 Error 0.1426667 8 0.0178333 Total 5.176 14 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Two-Way Factorial Design Cells . Column Treatment Row Treatment Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 43

Two-Way ANOVA: Hypotheses Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 44

Formulas for Computing a Two-Way ANOVA Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 45

A 2  3 Factorial Design with Interaction Cell Means C1 C2 C3 Row effects R1 R2 Column Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 46

A 2  3 Factorial Design with Some Interaction Cell Means C1 C2 C3 Row effects R1 R2 Column Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 47

A 2  3 Factorial Design with No Interaction Cell Means C1 C2 C3 Row effects R1 R2 Column Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 48

A 2  3 Factorial Design: Data and Measurements for CEO Dividend Example 1.75 2.75 3.625 Location Where Company Stock is Traded How Stockholders are Informed of Dividends NYSE AMEX OTC Annual/Quarterly Reports 2 1 3 4 2.5 Presentations to Analysts 2.9167 Xj Xi X11=1.5 X23=3.75 X22=3.0 X21=2.0 X13=3.5 X12=2.5 N = 24 n = 4 X=2.7083 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 49

A 2  3 Factorial Design: Calculations for the CEO Dividend Example (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 50

A 2  3 Factorial Design: Calculations for the CEO Dividend Example (Part 2) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

A 2  3 Factorial Design: Calculations for the CEO Dividend Example (Part 3) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 52

Analysis of Variance for the CEO Dividend Problem Source of Variance SS df MS F Row 1.0418 1 1.0418 2.42 Column 14.0833 2 7.0417 16.35* Interaction 0.0833 2 0.0417 0.10 Error 7.7500 18 0.4306 Total 22.9583 23 *Denotes significance at = .01. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 53

Excel Output for the CEO Dividend Example (Part 1) Anova: Two-Factor With Replication SUMMARY NYSE ASE OTC Total AQReport Count 4 12 Sum 6 10 14 30 Average 1.5 2.5 3.5 Variance 0.3333 1 Presentation 8 15 35 2 3 3.75 2.9167 0.6667 0.25 0.9924 22 29 1.75 2.75 3.625 0.5 0.2679 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Excel Output for the CEO Dividend Example (Part 2) ANOVA Source of Variation SS df MS F P-value F crit Sample 1.0417 1 2.4194 0.1373 4.4139 Columns 14.083 2 7.0417 16.355 9E-05 3.5546 Interaction 0.0833 0.0417 0.0968 0.9082 Within 7.75 18 0.4306 Total 22.958 23 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.