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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-1 Chapter 6 The Normal Distribution and Other Continuous Distributions Statistics for Managers Using Microsoft ® Excel 4 th Edition

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-2 Chapter Goals After completing this chapter, you should be able to: Describe the characteristics of the normal distribution Translate normal distribution problems into standardized normal distribution problems Find probabilities using a normal distribution table Evaluate the normality assumption

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-3 Chapter Goals After completing this chapter, you should be able to: Define the concept of a sampling distribution Determine the mean and standard deviation for the sampling distribution of the sample mean, X Determine the mean and standard deviation for the sampling distribution of the sample proportion, p s Describe the Central Limit Theorem and its importance Apply sampling distributions for both X and p s _ _ (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-4 Probability Distributions Continuous Probability Distributions Binomial Probability Distributions Discrete Probability Distributions Normal Uniform not covered Exponential not covered Ch. 5Ch. 6 Hypergeometric not covered Poisson not covered

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-5 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-6 The Normal Distribution Probability Distributions Normal Uniform not covered Exponential not covered Continuous Probability Distributions

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-7 The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to Mean = Median = Mode X f(X) μ σ

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-8 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-9 The Normal Distribution Shape X f(X) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-10 The Normal Probability Density Function The formula for the normal probability density function is Wheree = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 μ = the population mean σ = the population standard deviation X = any value of the continuous variable

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-11 The Standardized Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z) Need to transform X units into Z units

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-12 Translation to the Standardized Normal Distribution Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation: Z always has mean = 0 and standard deviation = 1

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-13 The Standardized Normal Probability Density Function The formula for the standardized normal probability density function is Wheree = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 Z = any value of the standardized normal distribution

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-14 The Standardized Normal Distribution Also known as the “Z” distribution Mean is 0 Standard Deviation is 1 Z f(Z) 0 1 Values above the mean have positive Z-values, values below the mean have negative Z-values

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-15 Example If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-16 Comparing X and Z units Z 100 2.00 200X Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z) (μ = 100, σ = 50) (μ = 0, σ = 1)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-17 Finding Normal Probabilities Probability is the area under the curve! ab X f(X) PaXb( ) ≤ Probability is measured by the area under the curve ≤ PaXb( ) << = (Note that the probability of any individual value is zero)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-18 f(X) X μ Probability as Area Under the Curve 0.5 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-19 Empirical Rules μ ± 1σ encloses about 68% of X’s f(X) X μμ+1σμ-1σ What can we say about the distribution of values around the mean? There are some general rules: σσ 68.26%

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-20 The Empirical Rule μ ± 2σ covers about 95% of X’s μ ± 3σ covers about 99.7% of X’s xμ 2σ2σ2σ2σ xμ 3σ3σ3σ3σ 95.44%99.72% (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-21 The Standardized Normal Table The Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z) Z 02.00.9772 Example: P(Z < 2.00) =.9772

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-22 The Standardized Normal Table The value within the table gives the probability from Z = up to the desired Z value.9772 2.0 P(Z < 2.00) =.9772 The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point 2.0...... (continued) Z 0.00 0.01 0.02 … 0.0 0.1

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-23 General Procedure for Finding Probabilities Draw the normal curve for the problem in terms of X Translate X-values to Z-values Use the Standardized Normal Table To find P(a < X < b) when X is distributed normally:

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-24 Finding Normal Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0 Find P(X < 8.6) X 8.6 8.0

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-25 Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6) Z 0.12 0 X 8.6 8 μ = 8 σ = 10 μ = 0 σ = 1 (continued) Finding Normal Probabilities P(X < 8.6)P(Z < 0.12)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-26 Z 0.12 Z.00.01 0.0.5000.5040.5080.5398.5438 0.2.5793.5832.5871 0.3.6179.6217.6255 Solution: Finding P(Z < 0.12).5478.02 0.1. 5478 Standardized Normal Probability Table (Portion) 0.00 = P(Z < 0.12) P(X < 8.6)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-27 Upper Tail Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(X > 8.6) X 8.6 8.0

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-28 Now Find P(X > 8.6)… (continued) Z 0.12 0 Z.5478 0 1.000 1.0 -.5478 =.4522 P(X > 8.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12) = 1.0 -.5478 =.4522 Upper Tail Probabilities

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-29 Probability Between Two Values Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(8 < X < 8.6) P(8 < X < 8.6) = P(0 < Z < 0.12) Z0.12 0 X8.6 8 Calculate Z-values:

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-30 Z 0.12 Solution: Finding P(0 < Z < 0.12).0478 0.00 = P(0 < Z < 0.12) P(8 < X < 8.6) = P(Z < 0.12) – P(Z ≤ 0) =.5478 -.5000 =.0478.5000 Z.00.01 0.0.5000.5040.5080.5398.5438 0.2.5793.5832.5871 0.3.6179.6217.6255.02 0.1. 5478 Standardized Normal Probability Table (Portion)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-31 Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < X < 8) X 7.4 8.0 Probabilities in the Lower Tail

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-32 Probabilities in the Lower Tail Now Find P(7.4 < X < 8)… X 7.48.0 P(7.4 < X < 8) = P(-0.12 < Z < 0) = P(Z < 0) – P(Z ≤ -0.12) =.5000 -.4522 =.0478 (continued).0478.4522 Z -0.12 0 The Normal distribution is symmetric, so this probability is the same as P(0 < Z < 0.12)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-33 Steps to find the X value for a known probability: 1. Find the Z value for the known probability 2. Convert to X units using the formula: Finding the X value for a Known Probability

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-34 Finding the X value for a Known Probability Example: Suppose X is normal with mean 8.0 and standard deviation 5.0. Now find the X value so that only 20% of all values are below this X X ?8.0.2000 Z ? 0 (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-35 Find the Z value for 20% in the Lower Tail 20% area in the lower tail is consistent with a Z value of -0.84 Z.03 -0.9.1762.1736.2033 -0.7.2327.2296.04 -0.8. 2005 Standardized Normal Probability Table (Portion).05.1711.1977.2266 … … … … X ?8.0.2000 Z -0.84 0 1. Find the Z value for the known probability

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-36 2. Convert to X units using the formula: Finding the X value So 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-37 Assessing Normality Not all continuous random variables are normally distributed It is important to evaluate how well the data set is approximated by a normal distribution

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-38 Assessing Normality Construct charts or graphs 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 σ? Is the range approximately 6 σ? (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-39 Assessing Normality Observe the distribution of the data set Do approximately 2/3 of the observations lie within mean 1 standard deviation? Do approximately 80% of the observations lie within mean 1.28 standard deviations? Do approximately 95% of the observations lie within mean 2 standard deviations? Evaluate normal probability plot Is the normal probability plot approximately linear with positive slope? (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-40 The Normal Probability Plot 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

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-41 A normal probability plot for data from a normal distribution will be approximately linear: 30 60 90 -2012 Z X The Normal Probability Plot (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-42 Normal Probability Plot Left-SkewedRight-Skewed Rectangular 30 60 90 -2 012 Z X (continued) 30 60 90 -2 012 Z X 30 60 90 -2 012 Z X Nonlinear plots indicate a deviation from normality

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-43 Sampling Distributions Sampling Distributions of the Mean Sampling Distributions of the Proportion

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-44 Sampling Distributions A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-45 Developing a Sampling Distribution Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) A B C D

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-46.3.2.1 0 18 20 22 24 A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution: Developing a Sampling Distribution

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-47 16 possible samples (sampling with replacement) Now consider all possible samples of size n=2 (continued) Developing a Sampling Distribution 16 Sample Means

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-48 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 a Sampling Distribution (continued) (no longer uniform) _

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-49 Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-50 Comparing the Population with its Sampling Distribution 18 19 20 21 22 23 24 0.1.2.3 P(X) X 18 20 22 24 A B C D 0.1.2.3 Population N = 4 P(X) X _ Sample Means Distribution n = 2 _

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-51 Sampling Distributions of the Mean Sampling Distributions Sampling Distributions of the Mean Sampling Distributions of the Proportion

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-52 Standard Error of the Mean Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: Note that the standard error of the mean decreases as the sample size increases

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-53 If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and (This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-54 Z-value for Sampling Distribution of the Mean Z-value for the sampling distribution of : where:= sample mean = population mean = population standard deviation n = sample size

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-55 Finite Population Correction Apply the Finite Population Correction if: the sample is large relative to the population (n is greater than 5% of N) and… Sampling is without replacement Then

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-56 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution Properties (i.e. is unbiased )

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-57 Sampling Distribution Properties For sampling with replacement: As n increases, decreases Larger sample size Smaller sample size (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-58 If the Population is not Normal We can apply the Central Limit Theorem: Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough. Properties of the sampling distribution: and

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-59 n↑n↑ Central Limit Theorem As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-60 Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation (Sampling with replacement) Larger sample size Smaller sample size If the Population is not Normal (continued) Sampling distribution properties:

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-61 How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normal. For very skewed distributions, even 30 might not be enough. For fairly symmetric distributions, n > 15, but it could be as low as n = 2! For normal population distributions, the sampling distribution of the mean is always normally distributed

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-62 Example Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2?

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-63 Example Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of is approximately normal … with mean = 8 …and standard deviation (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-64 Example Solution (continued): (continued) Z 7.8 8.2 -0.5 0.5 Sampling Distribution Standard Normal Distribution.1915 +.1915 Population Distribution ? ? ? ? ? ? ?? ? ? ? ? SampleStandardize X

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-65 Sampling Distributions of the Proportion Sampling Distributions Sampling Distributions of the Mean Sampling Distributions of the Proportion

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-66 Population Proportions, p p = the proportion of the population having some characteristic Sample proportion ( p s ) provides an estimate of p: 0 ≤ p s ≤ 1 p s has a binomial distribution (assuming sampling with replacement from a finite population or without replacement from an infinite population)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-67 Sampling Distribution of p Approximated by a normal distribution if: where and (where p = population proportion) Sampling Distribution P( p s ).3.2.1 0 0. 2.4.6 8 1 psps

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-68 Z-Value for Proportions If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor: Standardize p s to a Z value with the formula:

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-69 Example If the true proportion of voters who support Proposition A is p =.4, what is the probability that a sample of size 200 yields a sample proportion between.40 and.45? i.e.: if p =.4 and n = 200, what is P(.40 ≤ p s ≤.45) ?

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-70 Example if p =.4 and n = 200, what is P(.40 ≤ p s ≤.45) ? (continued) Find : Convert to standard normal:

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-71 Example Z.451.44.4251 Standardize Sampling Distribution Standardized Normal Distribution if p =.4 and n = 200, what is P(.40 ≤ p s ≤.45) ? (continued) Use standard normal table: P(0 ≤ Z ≤ 1.44) =.4251.400 psps

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-72 Chapter Summary Presented key continuous distributions normal, uniform, exponential Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 6-73 Chapter Summary Introduced sampling distributions Described the sampling distribution of the mean For normal populations Using the Central Limit Theorem Described the sampling distribution of a proportion Calculated probabilities using sampling distributions (continued)

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