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Statistics for Managers Using Microsoft Excel 3 rd Edition

Statistics for Managers Using Microsoft Excel 3 rd Edition. Chapter 5 The Normal Distribution and Sampling Distributions. Chapter Topics. The normal distribution The standardized normal distribution Evaluating the normality assumption The exponential distribution. Chapter Topics.

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Statistics for Managers Using Microsoft Excel 3 rd Edition

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  1. Statistics for Managers Using Microsoft Excel 3rd Edition Chapter 5 The Normal Distribution and Sampling Distributions

  2. Chapter Topics • The normal distribution • The standardized normal distribution • Evaluating the normality assumption • The exponential distribution

  3. Chapter Topics (continued) • Introduction to sampling distribution • Sampling distribution of the mean • Sampling distribution of the proportion • Sampling from finite population

  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

  5. The Normal Distribution • “Bell shaped” • Symmetrical • Mean, median and mode are equal • Interquartile rangeequals 1.33 s • Random variablehas infinite range f(X) X  Mean Median Mode

  6. The Mathematical Model

  7. Expectation

  8. Variance

  9. Many Normal Distributions There are an infinite number of normal distributions By varying the parameters  and , we obtain different normal distributions

  10. Finding Probabilities Probability is the area under the curve! f(X) X d c

  11. Which Table to Use? An infinite number of normal distributions means an infinite number of tables to look up!

  12. Solution: The Cumulative Standardized Normal Distribution Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5478 .5000 0.0 .5040 .5080 Shaded Area Exaggerated .5398 .5438 .5478 0.1 0.2 .5793 .5832 .5871 Probabilities Z = 0.12 0.3 .6179 .6217 .6255 Only One Table is Needed

  13. Standardizing Example Standardized Normal Distribution Normal Distribution Shaded Area Exaggerated

  14. Example: Standardized Normal Distribution Normal Distribution Shaded Area Exaggerated

  15. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5832 .5000 0.0 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.21 0.3 .6179 .6217 .6255

  16. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .4168 .3821 -03 .3783 .3745 Shaded Area Exaggerated -02 .4207 .4168 .4129 -0.1 .4602 .4562 .4522 Z = -0.21 0.0 .5000 .4960 .4920

  17. Normal Distribution in PHStat • PHStat | probability & prob. Distributions | normal … • Example in excel spreadsheet

  18. Example: Standardized Normal Distribution Normal Distribution Shaded Area Exaggerated

  19. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .6179 .5000 0.0 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.30 0.3 .6179 .6217 .6255

  20. Finding Z Values for Known Probabilities Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = 0.1217 ? .01 Z .00 0.2 0.0 .5040 .5000 .5080 .6217 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 .6179 .6255 .6217 0.3 Shaded Area Exaggerated

  21. Recovering X Values for Known Probabilities Standardized Normal Distribution Normal Distribution

  22. 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

  23. Assessing Normality (continued) • 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 s? • Is the range approximately 6 s?

  24. Assessing Normality (continued) • 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?

  25. 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

  26. Assessing Normality (continued) Normal Probability Plot for Normal Distribution 90 X 60 Z 30 -2 -1 0 1 2 Look for Straight Line!

  27. Normal Probability Plot Left-Skewed Right-Skewed 90 90 X X 60 60 Z Z 30 30 -2 -1 0 1 2 -2 -1 0 1 2 Rectangular U-Shaped 90 90 X X 60 60 Z Z 30 30 -2 -1 0 1 2 -2 -1 0 1 2

  28. Exponential Distributions e.g.: Drivers Arriving at a Toll Bridge; Customers Arriving at an ATM Machine

  29. Exponential Distributions (continued) • Describes time or distance between events • Used for queues • Density function • Parameters f(X)  = 0.5  = 2.0 X

  30. 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?

  31. Exponential Distribution in PHStat • PHStat | probability & prob. Distributions | exponential • Example in excel spreadsheet

  32. 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

  33. 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

  34. 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 inyears C B D A

  35. Developing Sampling Distributions (continued) Summary Measures for the Population Distribution P(X) .3 .2 .1 0 X A B C D (18) (20) (22) (24) Uniform Distribution

  36. All Possible Samples of Size n=2 Developing Sampling Distributions (continued) 16 Sample Means 16 Samples Taken with Replacement

  37. Sampling Distribution of All Sample Means Developing Sampling Distributions (continued) Sample Means Distribution 16 Sample Means P(X) .3 .2 .1 _ 0 X 18 19 20 21 22 23 24

  38. Summary Measures of Sampling Distribution Developing Sampling Distributions (continued)

  39. Comparing the Population with its Sampling Distribution Population N = 4 Sample Means Distribution n = 2 P(X) P(X) .3 .3 .2 .2 .1 .1 _ 0 0 X AB C D (18)(20)(22)(24) 18 19 20 21 22 23 24 X

  40. 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

  41. Unbiasedness P(X) Unbiased Biased

  42. Less Variability P(X) Sampling Distribution of Median Sampling Distribution of Mean

  43. Effect of Large Sample Larger sample size P(X) Smaller sample size

  44. When the Population is Normal Population Distribution Central Tendency Variation Sampling Distributions Sampling with Replacement

  45. When the Population is Not Normal Population Distribution Central Tendency Variation Sampling Distributions Sampling with Replacement

  46. Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As sample size gets large enough…

  47. 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

  48. Example: Standardized Normal Distribution Sampling Distribution

  49. 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

  50. Sampling Distribution of Sample Proportion • Approximated by normal distribution • Mean: • Standard error: Sampling Distribution P(ps) .3 .2 .1 0 ps 0 . 2 .4 .6 8 1 p = population proportion

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