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

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Chapter Topics The normal distribution The standardized normal distribution Evaluating the normality assumption The exponential distribution

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

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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

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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)

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The Mathematical Model

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Expectation

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Variance

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

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

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

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Solution: The Cumulative Standardized Normal Distribution Z.00.01 0.0.5000.5040.5080.5398.5438 0.2.5793.5832.5871 0.3.6179.6217.6255.5478.02 0.1. 5478 Cumulative Standardized Normal Distribution Table (Portion) Probabilities Shaded Area Exaggerated Only One Table is Needed Z = 0.12

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Standardizing Example Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

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Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

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Z.00.01 0.0.5000.5040.5080.5398.5438 0.2.5793.5832.5871 0.3.6179.6217.6255.5832.02 0.1. 5478 Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.21 Example: (continued)

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Z.00.01 -03.3821.3783.3745.4207.4168 -0.1.4602.4562.4522 0.0.5000.4960.4920.4168.02 -02.4129 Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = -0.21 Example: (continued)

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

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Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

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Example: (continued) Z.00.01 0.0.5000.5040.5080.5398.5438 0.2.5793.5832.5871 0.3.6179.6217.6255.6179.02 0.1. 5478 Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.30

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.6217 Finding Z Values for Known Probabilities Z.000.2 0.0.5000.5040.5080 0.1.5398.5438.5478 0.2.5793.5832.5871.6179.6255.01 0.3 Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = 0.1217 ? Shaded Area Exaggerated.6217

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Recovering X Values for Known Probabilities Normal Distribution Standardized Normal Distribution

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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

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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)

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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)

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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)

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Assessing Normality Normal Probability Plot for Normal Distribution Look for Straight Line! 30 60 90 -2012 Z X (continued)

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Normal Probability Plot Left-SkewedRight-Skewed RectangularU-Shaped 30 60 90 -2012 Z X 30 60 90 -2012 Z X 30 60 90 -2012 Z X 30 60 90 -2012 Z X

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

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

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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?

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

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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

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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

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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

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.3.2.1 0 A B C D (18) (20) (22) (24) Uniform Distribution P(X) X Developing Sampling Distributions (continued) Summary Measures for the Population Distribution

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

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Sampling Distribution of All Sample Means 18 19 20 21 22 23 24 0.1.2.3 P(X) X Sample Means Distribution 16 Sample Means _ Developing Sampling Distributions (continued)

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Summary Measures of Sampling Distribution Developing Sampling Distributions (continued)

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Comparing the Population with its Sampling Distribution 18 19 20 21 22 23 24 0.1.2.3 P(X) X Sample Means Distribution n = 2 A B C D (18) (20) (22) (24) 0.1.2.3 Population N = 4 P(X) X _

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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

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Unbiasedness BiasedUnbiased P(X)

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

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

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

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

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

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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

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Example: Sampling Distribution Standardized Normal Distribution

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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

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Sampling Distribution of Sample Proportion Approximated by normal distribution Mean: Standard error: p = population proportion Sampling Distribution P(p s ).3.2.1 0 0. 2.4.6 8 1 psps

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Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution

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Example: Sampling Distribution Standardized Normal Distribution

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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

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Applied Quantitative Analysis and Practices

Applied Quantitative Analysis and Practices

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