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Statistics for Managers Using Microsoft Excel 3rd Edition Chapter 5 The Normal Distribution and Sampling Distributions © 2002 Prentice-Hall, Inc. Chap 5-1 Chapter Topics The normal distribution The standardized normal distribution Evaluating the normality assumption The exponential distribution © 2002 Prentice-Hall, Inc. Chap 5-2 Chapter Topics Introduction to sampling distribution Sampling distribution of the mean Sampling distribution of the proportion Sampling from finite population © 2002 Prentice-Hall, Inc. (continued) Chap 5-3 Continuous Probability Distributions Continuous random variable Continuous probability distribution Values from interval of numbers Absence of gaps Distribution of continuous random variable Most important continuous probability distribution The normal distribution © 2002 Prentice-Hall, Inc. Chap 5-4 The Normal Distribution “Bell shaped” Symmetrical Mean, median and mode are equal Interquartile range equals 1.33 s Random variable has infinite range © 2002 Prentice-Hall, Inc. f(X) X Mean Median Mode Chap 5-5 The Mathematical Model f X 1 e 1 2s 2 X 2s 2 f X : density of random variable X 3.14159; e 2.71828 : population mean s : population standard deviation X : value of random variable X © 2002 Prentice-Hall, Inc. Chap 5-6 Many Normal Distributions There are an infinite number of normal distributions By varying the parameters s and , we obtain different normal distributions © 2002 Prentice-Hall, Inc. Chap 5-7 Finding Probabilities Probability is the area under the curve! P c X d ? f(X) c © 2002 Prentice-Hall, Inc. d X Chap 5-8 Which Table to Use? An infinite number of normal distributions means an infinite number of tables to look up! © 2002 Prentice-Hall, Inc. Chap 5-9 Solution: The Cumulative Standardized Normal Distribution Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 sZ 1 .02 .5478 0.0 .5000 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Probabilities 0.3 .6179 .6217 .6255 © 2002 Prentice-Hall, Inc. 0 Z = 0.12 Only One Table is Needed Chap 5-10 Standardizing Example Z X s 6.2 5 0.12 10 Standardized Normal Distribution Normal Distribution s 10 5 © 2002 Prentice-Hall, Inc. sZ 1 6.2 X Shaded Area Exaggerated Z 0 0.12 Z Chap 5-11 Example: P 2.9 X 7.1 .1664 Z X s 2.9 5 .21 10 Z X s 7.1 5 .21 10 Standardized Normal Distribution Normal Distribution s 10 .0832 sZ 1 .0832 2.9 5 © 2002 Prentice-Hall, Inc. 7.1 X 0.21 Shaded Area Exaggerated Z 0 0.21 Z Chap 5-12 Example: P 2.9 X 7.1 .1664(continued) Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 sZ 1 .02 .5832 0.0 .5000 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 © 2002 Prentice-Hall, Inc. 0 Z = 0.21 Chap 5-13 Example: P 2.9 X 7.1 .1664(continued) Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 .02 Z 0 sZ 1 .4168 -03 .3821 .3783 .3745 Shaded Area Exaggerated -02 .4207 .4168 .4129 -0.1 .4602 .4562 .4522 0.0 .5000 .4960 .4920 © 2002 Prentice-Hall, Inc. 0 Z = -0.21 Chap 5-14 Normal Distribution in PHStat PHStat | probability & prob. Distributions | normal … Example in excel spreadsheet © 2002 Prentice-Hall, Inc. Chap 5-15 Example: P X 8 .3821 Z X s 85 .30 10 Standardized Normal Distribution Normal Distribution s 10 sZ 1 .3821 5 © 2002 Prentice-Hall, Inc. 8 X Shaded Area Exaggerated Z 0 0.30 Z Chap 5-16 Example: P X 8 .3821 Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 (continued) sZ 1 .02 .6179 0.0 .5000 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 © 2002 Prentice-Hall, Inc. 0 Z = 0.30 Chap 5-17 Finding Z Values for Known Probabilities What is Z Given Probability = 0.1217 ? Z 0 sZ 1 Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 0.2 0.0 .5000 .5040 .5080 .6217 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Shaded Area Exaggerated © 2002 Prentice-Hall, Inc. 0 Z .31 0.3 .6179 .6217 .6255 Chap 5-18 Recovering X Values for Known Probabilities Standardized Normal Distribution Normal Distribution s 10 sZ 1 .1179 .3821 5 ? X Z 0 0.30 Z X Zs 5 .3010 8 © 2002 Prentice-Hall, Inc. Chap 5-19 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 5-20 Assessing Normality Construct charts (continued) For small- or moderate-sized data sets, do stemand-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 s? Is the range approximately 6 s? © 2002 Prentice-Hall, Inc. Chap 5-21 Assessing Normality Observe the distribution of the data set (continued) Do approximately between mean Do approximately between mean Do approximately between mean 2/3 of the observations lie 1 standard deviation? 4/5 of the observations lie 1.28 standard deviations? 19/20 of the observations lie 2 standard deviations? Evaluate normal probability plot Do the points lie on or close to a straight line with positive slope? © 2002 Prentice-Hall, Inc. Chap 5-22 Assessing Normality (continued) 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 © 2002 Prentice-Hall, Inc. Chap 5-23 Assessing Normality (continued) Normal Probability Plot for Normal Distribution 90 X 60 Z 30 -2 -1 0 1 2 © 2002 Prentice-Hall, Inc. Look for Straight Line! Chap 5-24 Normal Probability Plot Left-Skewed Right-Skewed 90 90 X 60 X 60 Z 30 -2 -1 0 1 2 -2 -1 0 1 2 Rectangular U-Shaped 90 90 X 60 X 60 Z 30 -2 -1 0 1 2 © 2002 Prentice-Hall, Inc. Z 30 Z 30 -2 -1 0 1 2 Chap 5-25 Exponential Distributions P arrival time X 1 e X X : any value of continuous random variable : the population average number of arrivals per unit of time 1/: average time between arrivals e 2.71828 e.g.: Drivers Arriving at a Toll Bridge; Customers Arriving at an ATM Machine © 2002 Prentice-Hall, Inc. Chap 5-26 Exponential Distributions (continued) Describes time or distance between events f(X) Density function Used for queues f x Parameters © 2002 Prentice-Hall, Inc. 1 e x = 0.5 = 2.0 X s Chap 5-27 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? 30 X 5 / 60 hours P arrival time >X 1 P arrival time X 1 1 e 30 5/ 60 .0821 © 2002 Prentice-Hall, Inc. Chap 5-28 Exponential Distribution in PHStat PHStat | probability & prob. Distributions | exponential Example in excel spreadsheet © 2002 Prentice-Hall, Inc. Chap 5-29 Why Study Sampling Distributions Sample statistics are used to estimate population parameters e.g.: X 50 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 5-30 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 5-31 Developing Sampling Distributions Assume there is a population … Population size N=4 B C Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 measured in years A © 2002 Prentice-Hall, Inc. D Chap 5-32 Developing Sampling Distributions (continued) Summary Measures for the Population Distribution N X i 1 P(X) i .3 N 18 20 22 24 21 4 N s X i 1 © 2002 Prentice-Hall, Inc. i N .2 .1 0 2 2.236 A B C D (18) (20) (22) (24) X Uniform Distribution Chap 5-33 Developing Sampling Distributions All Possible Samples of Size n=2 1st Obs 2nd Observation 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 (continued) 16 Sample Means 22 22,18 22,20 22,22 22,24 1st 2nd Observation Obs 18 20 22 24 24 24,18 24,20 24,22 24,24 18 18 19 20 21 16 Samples Taken with Replacement 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 © 2002 Prentice-Hall, Inc. Chap 5-34 Developing Sampling Distributions (continued) Sampling Distribution of All Sample Means Sample Means Distribution 16 Sample Means 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 © 2002 Prentice-Hall, Inc. P(X) .3 .2 .1 0 _ 18 19 20 21 22 23 24 X Chap 5-35 Developing Sampling Distributions (continued) Summary Measures of Sampling Distribution N X X i 1 N i 18 19 19 16 N sX X i 1 i X © 2002 Prentice-Hall, Inc. 21 2 N 18 21 19 21 2 24 16 2 24 21 2 1.58 Chap 5-36 Comparing the Population with its Sampling Distribution Population N=4 21 s 2.236 Sample Means Distribution n=2 X 21 P(X) .3 P(X) .3 .2 .2 .1 .1 0 0 A B C (18) (20) (22) © 2002 Prentice-Hall, Inc. D X s X 1.58 _ 18 19 20 21 22 23 24 X (24) Chap 5-37 Properties of Summary Measures X I.E. X Is unbiased Standard error (standard deviation) of the sampling distribution s X is less than the standard error of other unbiased estimators For sampling with replacement: © 2002 Prentice-Hall, Inc. As n increases, sX decreases sX s n Chap 5-38 Unbiasedness P(X) Unbiased © 2002 Prentice-Hall, Inc. Biased X X Chap 5-39 Less Variability P(X) Sampling Distribution of Median Sampling Distribution of Mean © 2002 Prentice-Hall, Inc. X Chap 5-40 Effect of Large Sample Larger sample size P(X) Smaller sample size © 2002 Prentice-Hall, Inc. X Chap 5-41 When the Population is Normal Population Distribution Central Tendency X Variation sX s n Sampling with Replacement © 2002 Prentice-Hall, Inc. s 10 50 Sampling Distributions n4 n 16 sX 5 s X 2.5 X 50 X Chap 5-42 When the Population is Not Normal Population Distribution Central Tendency X Variation sX s n Sampling with Replacement © 2002 Prentice-Hall, Inc. s 10 50 Sampling Distributions n4 n 30 sX 5 s X 1.8 X 50 X Chap 5-43 Central Limit Theorem As sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population X © 2002 Prentice-Hall, Inc. Chap 5-44 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 5-45 Example: 8 s =2 n 25 P 7.8 X 8.2 ? 7.8 8 X X 8.2 8 P 7.8 X 8.2 P sX 2 / 25 2 / 25 P .5 Z .5 .3830 Standardized Normal Distribution Sampling Distribution 2 sX .4 25 sZ 1 .1915 7.8 © 2002 Prentice-Hall, Inc. 8.2 X 8 X 0.5 Z 0 0.5 Z Chap 5-46 Population Proportions Categorical variable e.g.: Gender, voted for Bush, college degree Proportion of population having a characteristic p Sample proportion provides an estimate X number of successes pS n sample size p If two outcomes, X has a binomial distribution Possess or do not possess characteristic © 2002 Prentice-Hall, Inc. Chap 5-47 Sampling Distribution of Sample Proportion Approximated by normal distribution np 5 n 1 p 5 P(ps) .3 .2 .1 0 Mean: p p Sampling Distribution 0 .2 .4 .6 8 1 ps S Standard error: sp S © 2002 Prentice-Hall, Inc. p 1 p n p = population proportion Chap 5-48 Standardizing Sampling Distribution of Proportion Z pS pS sp S p 1 p n Standardized Normal Distribution Sampling Distribution sp pS p sZ 1 S p © 2002 Prentice-Hall, Inc. S pS Z 0 Z Chap 5-49 Example: n 200 p .4 P pS .43 ? p .43 .4 S pS P pS .43 P s pS .4 1 .4 200 Standardized Normal Distribution Sampling Distribution sp sZ 1 S © 2002 Prentice-Hall, Inc. P Z .87 .8078 p .43 S pS 0 .87 Z Chap 5-50 Sampling from Finite Sample Modify standard error if sample size (n) is large relative to population size (N ) n .05N or n / N .05 Use finite population correction factor (fpc) Standard error with FPC sX sP S © 2002 Prentice-Hall, Inc. s n N n N 1 p 1 p N n n N 1 Chap 5-51 Chapter Summary Discussed the normal distribution Described the standard normal distribution Evaluated the normality assumption Defined the exponential distribution © 2002 Prentice-Hall, Inc. Chap 5-52 Chapter Summary (continued) Introduced sampling distributions Discussed sampling distribution of the sample mean Described sampling distribution of the sample proportion Discussed sampling from finite populations © 2002 Prentice-Hall, Inc. Chap 5-53

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