© 2002 Prentice-Hall, Inc.Chap 5-1 Statistics for Managers Using Microsoft Excel 3 rd Edition Chapter 5 The Normal Distribution and Sampling Distributions

© 2002 Prentice-Hall, Inc. Chap 5-2 Chapter Topics The normal distribution The standardized normal distribution Evaluating the normality assumption The exponential distribution

© 2002 Prentice-Hall, Inc. Chap 5-3 Chapter Topics Introduction to sampling distribution Sampling distribution of the mean Sampling distribution of the proportion Sampling from finite population (continued)

© 2002 Prentice-Hall, Inc. Chap 5-4 Continuous Probability Distributions Continuous random variable Values from interval of numbers Absence of gaps Continuous probability distribution Distribution of continuous random variable Most important continuous probability distribution The normal distribution

© 2002 Prentice-Hall, Inc. Chap 5-5 The Normal Distribution “Bell shaped” Symmetrical Mean, median and mode are equal Interquartile range equals 1.33  Random variable has infinite range Mean Median Mode X f(X) 

© 2002 Prentice-Hall, Inc. Chap 5-6 The Mathematical Model

© 2002 Prentice-Hall, Inc. Chap 5-7 Many Normal Distributions By varying the parameters  and , we obtain different normal distributions There are an infinite number of normal distributions

© 2002 Prentice-Hall, Inc. Chap 5-8 Finding Probabilities Probability is the area under the curve! c d X f(X)f(X)

© 2002 Prentice-Hall, Inc. Chap 5-9 Which Table to Use? An infinite number of normal distributions means an infinite number of tables to look up!

© 2002 Prentice-Hall, Inc. Chap 5-10 Solution: The Cumulative Standardized Normal Distribution Z Cumulative Standardized Normal Distribution Table (Portion) Probabilities Shaded Area Exaggerated Only One Table is Needed Z = 0.12

© 2002 Prentice-Hall, Inc. Chap 5-11 Standardizing Example Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

© 2002 Prentice-Hall, Inc. Chap 5-12 Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

© 2002 Prentice-Hall, Inc. Chap 5-13 Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.21 Example: (continued)

© 2002 Prentice-Hall, Inc. Chap 5-14 Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = Example: (continued)

© 2002 Prentice-Hall, Inc. Chap 5-15 Normal Distribution in PHStat PHStat | probability & prob. Distributions | normal … Example in excel spreadsheet

© 2002 Prentice-Hall, Inc. Chap 5-16 Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

© 2002 Prentice-Hall, Inc. Chap 5-17 Example: (continued) Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.30

© 2002 Prentice-Hall, Inc. Chap Finding Z Values for Known Probabilities Z Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = ? Shaded Area Exaggerated.6217

© 2002 Prentice-Hall, Inc. Chap 5-19 Recovering X Values for Known Probabilities Normal Distribution Standardized Normal Distribution

© 2002 Prentice-Hall, Inc. Chap 5-20 Assessing Normality Not all continuous random variables are normally distributed It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution

© 2002 Prentice-Hall, Inc. Chap 5-21 Assessing Normality Construct charts For small- or moderate-sized data sets, do stem- and-leaf display and box-and-whisker plot look symmetric? For large data sets, does the histogram or polygon appear bell-shaped? Compute descriptive summary measures Do the mean, median and mode have similar values? Is the interquartile range approximately 1.33  ? Is the range approximately 6  ? (continued)

© 2002 Prentice-Hall, Inc. Chap 5-22 Assessing Normality Observe the distribution of the data set Do approximately 2/3 of the observations lie between mean 1 standard deviation? Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? Do approximately 19/20 of the observations lie between mean 2 standard deviations? Evaluate normal probability plot Do the points lie on or close to a straight line with positive slope? (continued)

© 2002 Prentice-Hall, Inc. Chap 5-23 Assessing Normality Normal probability plot Arrange data into ordered array Find corresponding standardized normal quantile values Plot the pairs of points with observed data values on the vertical axis and the standardized normal quantile values on the horizontal axis Evaluate the plot for evidence of linearity (continued)

© 2002 Prentice-Hall, Inc. Chap 5-24 Assessing Normality Normal Probability Plot for Normal Distribution Look for Straight Line! Z X (continued)

© 2002 Prentice-Hall, Inc. Chap 5-25 Normal Probability Plot Left-SkewedRight-Skewed RectangularU-Shaped Z X Z X Z X Z X

© 2002 Prentice-Hall, Inc. Chap 5-26 Exponential Distributions e.g.: Drivers Arriving at a Toll Bridge; Customers Arriving at an ATM Machine

© 2002 Prentice-Hall, Inc. Chap 5-27 Exponential Distributions Describes time or distance between events Used for queues Density function Parameters (continued) f(X) X = 0.5 = 2.0

© 2002 Prentice-Hall, Inc. Chap 5-28 Example e.g.: Customers arrive at the check out line of a supermarket at the rate of 30 per hour. What is the probability that the arrival time between consecutive customers to be greater than five minutes?

© 2002 Prentice-Hall, Inc. Chap 5-29 Exponential Distribution in PHStat PHStat | probability & prob. Distributions | exponential Example in excel spreadsheet

© 2002 Prentice-Hall, Inc. Chap 5-30 Why Study Sampling Distributions Sample statistics are used to estimate population parameters e.g.: Estimates the population mean Problems: different samples provide different estimate Large samples gives better estimate; Large samples costs more How good is the estimate? Approach to solution: theoretical basis is sampling distribution

© 2002 Prentice-Hall, Inc. Chap 5-31 Sampling Distribution Theoretical probability distribution of a sample statistic Sample statistic is a random variable Sample mean, sample proportion Results from taking all possible samples of the same size

© 2002 Prentice-Hall, Inc. Chap 5-32 Developing Sampling Distributions Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 measured in years A B C D

© 2002 Prentice-Hall, Inc. Chap A B C D (18) (20) (22) (24) Uniform Distribution P(X) X Developing Sampling Distributions (continued) Summary Measures for the Population Distribution

© 2002 Prentice-Hall, Inc. Chap 5-34 All Possible Samples of Size n=2 16 Samples Taken with Replacement 16 Sample Means Developing Sampling Distributions (continued)

© 2002 Prentice-Hall, Inc. Chap 5-35 Sampling Distribution of All Sample Means P(X) X Sample Means Distribution 16 Sample Means _ Developing Sampling Distributions (continued)

© 2002 Prentice-Hall, Inc. Chap 5-36 Summary Measures of Sampling Distribution Developing Sampling Distributions (continued)

© 2002 Prentice-Hall, Inc. Chap 5-37 Comparing the Population with its Sampling Distribution P(X) X Sample Means Distribution n = 2 A B C D (18) (20) (22) (24) Population N = 4 P(X) X _

© 2002 Prentice-Hall, Inc. Chap 5-38 Properties of Summary Measures I.E. Is unbiased Standard error (standard deviation) of the sampling distribution is less than the standard error of other unbiased estimators For sampling with replacement: As n increases, decreases

© 2002 Prentice-Hall, Inc. Chap 5-39 Unbiasedness BiasedUnbiased P(X)

© 2002 Prentice-Hall, Inc. Chap 5-40 Less Variability Sampling Distribution of Median Sampling Distribution of Mean P(X)

© 2002 Prentice-Hall, Inc. Chap 5-41 Effect of Large Sample Larger sample size Smaller sample size P(X)

© 2002 Prentice-Hall, Inc. Chap 5-42 When the Population is Normal Central Tendency Variation Sampling with Replacement Population Distribution Sampling Distributions

© 2002 Prentice-Hall, Inc. Chap 5-43 When the Population is Not Normal Central Tendency Variation Sampling with Replacement Population Distribution Sampling Distributions

© 2002 Prentice-Hall, Inc. Chap 5-44 Central Limit Theorem As sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population

© 2002 Prentice-Hall, Inc. Chap 5-45 How Large is Large Enough? For most distributions, n>30 For fairly symmetric distributions, n>15 For normal distribution, the sampling distribution of the mean is always normally distributed

© 2002 Prentice-Hall, Inc. Chap 5-46 Example: Sampling Distribution Standardized Normal Distribution

© 2002 Prentice-Hall, Inc. Chap 5-47 Population Proportions Categorical variable e.g.: Gender, voted for Bush, college degree Proportion of population having a characteristic Sample proportion provides an estimate If two outcomes, X has a binomial distribution Possess or do not possess characteristic

© 2002 Prentice-Hall, Inc. Chap 5-48 Sampling Distribution of Sample Proportion Approximated by normal distribution Mean: Standard error: p = population proportion Sampling Distribution P(p s ) psps

© 2002 Prentice-Hall, Inc. Chap 5-49 Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution

© 2002 Prentice-Hall, Inc. Chap 5-50 Example: Sampling Distribution Standardized Normal Distribution

© 2002 Prentice-Hall, Inc. Chap 5-51 Sampling from Finite Sample Modify standard error if sample size (n) is large relative to population size (N ) Use finite population correction factor (fpc) Standard error with FPC

© 2002 Prentice-Hall, Inc. Chap 5-52 Chapter Summary Discussed the normal distribution Described the standard normal distribution Evaluated the normality assumption Defined the exponential distribution

© 2002 Prentice-Hall, Inc. Chap 5-53 Chapter Summary Introduced sampling distributions Discussed sampling distribution of the sample mean Described sampling distribution of the sample proportion Discussed sampling from finite populations (continued)