Business Statistics, 4th by Ken Black

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 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

Process of Inferential Statistics

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

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

Relative Frequency Relative Class Interval Frequency Frequency
20-under 30-under 40-under 50-under 60-under 70-under Total Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Cumulative Frequency Cumulative Class Interval Frequency Frequency
20-under 30-under 40-under 50-under 60-under 70-under 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-under 40-under 50-under 60-under 70-under Total 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-under 40-under 50-under 60-under 70-under Total 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-under 40-under 50-under 60-under 70-under 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

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 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 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.

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 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 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

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 th 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

Variability No Variability in Cash Flow Variability in Cash Flow Mean

Variability Variability No Variability 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 = = 13 35 37 39 40 41 43 44 45 46 48 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

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

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

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-under 40-under 50-under 60-under 70-under Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40

Median of Grouped Data -- Example
Cumulative Class Interval Frequency Frequency 20-under 30-under 40-under 50-under 60-under 70-under 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-under 50-under 60-under 70 3 70-under 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

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

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 = Q IQR Upper inner fence = Q IQR Outer Fences Lower outer fence = Q IQR Upper outer fence = Q IQR Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 52

Box and Whisker Plot Q1 Q3 Q2 Minimum Maximum 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

Three Degrees of Correlation

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 2,725 619,207 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)

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 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

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

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

Venn Diagram of the X or Y but not Both Case

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.

Law of Multiplication Demonstration Problem 4.5

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

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 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 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

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

(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

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: Demonstration Problem 5.3

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

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: 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 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 5 0.0141 0.0131 6 0.0035 0.0030 7 0.0008 0.0006 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 1 0.3426 0.3292 0.0133 2 0.3689 0.0397 3 0.1581 0.1646 4 0.0264 0.0412 5 0.0013 0.0041 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 1 0.3306 0.3292 0.0014 2 0.3327 0.0035 3 0.1642 0.1646 4 0.0398 0.0412 5 0.0038 0.0041 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.

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

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

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

Table Lookup of a Standard Normal Probability
-3 -2 -1 1 2 3 Z 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

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

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
years old (homogeneous within) (alike) years old 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

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

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

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

Solution to Tire Store Example

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, % 0.998 6, % 0.992 6, % 0.958 2, % 0.993 2, % 0.975 2, % 0.866 % 0.971 % 0.950 % 0.895 % 0.924 % 0.868 % 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

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 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 

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

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

Confidence Interval to Estimate  when n is Large and  is Known

Car Rental Firm Example

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 = Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Confidence Intervals for  of a Normal Population: Unknown 

Comp Time: Excel Normal View

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

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

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 E-04 E-03 2 3 4 5 6 7 8 9 10 20 21 22 23 24 25 70 80 90 100 With df = 5 and a = 0.10, c2 = 5 10 15 20 0.10 df = 5 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 df 0.950 0.050 1 E-03 2 3 4 5 6 7 8 9 10 20 21 22 23 24 25 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

90% Confidence Interval for 2

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

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

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

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)

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 = vs mu < 4.300 The assumed sigma = 0.574 Variable N MEAN STDEV SE MEAN Z P VALUE Ratings Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Demonstration Problem 9.1: Excel (Part 1)

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 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)

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

Machine Plate Example: Excel (Part 1)

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 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

z Test of Population Proportion

Testing Hypotheses about a Proportion: Manufacturer Example (Part 1)

Testing Hypotheses about a Proportion: Manufacturer Example (Part 2)

Hypothesis Test for 2: Demonstration Problem 9.4 (Part 1)
df = 15 .05 .95 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 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

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

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

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 - = Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

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 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)

Difference Between Means: Using Excel
z-Test: Two Sample for Means Adv Mgr Auditing Mgr Mean 62.187 Known Variance 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

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

Hernandez Manufacturing Company (part 1)

MINITAB Output for Hernandez New-Employee Training Problem
Twosample T for method A vs method B N Mean StDev SE Mean method A method B 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= 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

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

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

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.

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

Confidence Intervals for Mean Difference for Related Samples

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

Sampling Distribution of Differences in Sample Proportions

Z Formula for the Difference in Two Population Proportions

Z Formula to Test the Difference in Population Proportions

Testing the Difference in Population Proportions (Demonstration Problem 10.6)

Confidence Interval to Estimate p1 - p2

Example Problem: When do men shop for groceries?

F Test for Two Population Variances

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)

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

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

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.

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

One-Way ANOVA: Sums of Squares Definitions

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

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 = nj n1 = 5 n2 = 8 n3 = 7 n4 = 4 N = 24 Mean Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

One-Way ANOVA: Sum of Squares Calculations

One-Way ANOVA: Mean Square and F Calculations

Analysis of Variance for Valve Openings
Source of Variance df SS MS F Between Error Total 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 Operator 2 8 50.22 6.2775 Operator 3 7 45.42 Operator 4 4 24.92 6.23 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 3 Within Groups 20 Total 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

Data from Demonstration Problem 11.1
PLANT (Employee Age) Group Means nj 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

Tukey-Kramer Procedure: The Case of Unequal Sample Sizes

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

Randomized Block Design: Computational Formulas

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)

Randomized Block Design: Sum of Squares Calculations (Part 2)

Randomized Block Design: Mean Square Calculations

Analysis of Variance for the Tread-Wear Example
Source of Variance SS df MS F Treatment Block Error Total Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Randomized Block Design Treatment Effects: Procedural Summary

Randomized Block Design Blocking Effects: Procedural Overview

Excel Output for Tread-Wear Example: Randomized Block Design
Anova: Two-Factor Without Replication SUMMARY Count Sum Average Variance Suplier 1 3 11.3 Suplier 2 10.1 Suplier 3 10.6 Suplier 4 9.3 3.1 0.21 Suplier 5 12.1 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 4 Columns 3.484 2 1.742 2.395E-06 Error 8 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

Formulas for Computing a Two-Way ANOVA

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

A 2  3 Factorial Design with No Interaction

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)

Analysis of Variance for the CEO Dividend Problem
Source of Variance SS df MS F Row Column * Interaction Error Total *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.

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