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Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Statistical Inferences Based on Two Samples Chapter 10

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10-2 Chapter Outline 10.1Comparing Two Population Means by Using Independent Samples: Variances Known 10.2Comparing Two Population Means by Using Independent Samples: Variances Unknown 10.3Paired Difference Experiments 10.4Comparing Two Population Proportions by Using Large, Independent Samples 10.5Comparing Two Population Variances by Using Independent Samples

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Comparing Two Population Means by Using Independent Samples: Variances Known Suppose a random sample has been taken from each of two different populations Suppose that the populations are independent of each other Then the random samples are independent of each other Then the sampling distribution of the difference in sample means is normally distributed

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10-4 Sampling Distribution of the Difference of Two Sample Means #1 Suppose population 1 has mean µ 1 and variance σ 1 2 From population 1, a random sample of size n 1 is selected which has mean 1 and variance s 1 2 Suppose population 2 has mean µ 2 and variance σ 2 2 From population 2, a random sample of size n 2 is selected which has mean 2 and variance s 2 2 Then the sample distribution of the difference of two sample means…

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10-5 Sampling Distribution of the Difference of Two Sample Means #2 Is normal, if each of the sampled populations is normal Approximately normal if the sample sizes n 1 and n 2 are large Has mean µ 1– 2 = µ 1 – µ 2 Has standard deviation

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10-6 Sampling Distribution of the Difference of Two Sample Means #3 Figure 10.1

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10-7 z-Based Confidence Interval for the Difference in Means (Variances Known) A 100(1 – ) percent confidence interval for the difference in populations µ 1 – µ 2 is

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10-8 z-Based Test About the Difference in Means (Variances Known) Test the null hypothesis about H 0 : µ 1 – µ 2 = D 0 D 0 = µ 1 – µ 2 is the claimed difference between the population means D 0 is a number whose value varies depending on the situation Often D 0 = 0, and the null means that there is no difference between the population means

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10-9 z-Based Test About the Difference in Means (Variances Known) Use the notation from the confidence interval statement on a prior slide Assume that each sampled population is normal or that the samples sizes n 1 and n 2 are large

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10-10 Test Statistic (Variances Known) The test statistic is The sampling distribution of this statistic is a standard normal distribution If the populations are normal and the samples are independent...

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10-11 z-Based Test About the Difference in Means (Variances Known) Reject H 0 : µ 1 – µ 2 = D 0 in favor of a particular alternative hypothesis at a level of significance if the appropriate rejection point rule holds or if the corresponding p-value is less than Rules are on the next slide…

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10-12 z-Based Test About the Difference in Means (Variances Known) Continued

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10-13 Example 10.2: The Bank Customer Waiting Time Case

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Comparing Two Population Means by Using Independent Samples: Variances Unknown Generally, the true values of the population variances σ 1 2 and σ 2 2 are not known They have to be estimated from the sample variances s 1 2 and s 2 2, respectively

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10-15 Comparing Two Population Means Continued Also need to estimate the standard deviation of the sampling distribution of the difference between sample means Two approaches: 1. If it can be assumed that σ 1 2 = σ 2 2 = σ 2, then calculate the “pooled estimate” of σ 2 2. If σ 1 2 ≠ σ 2 2, then use approximate methods

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10-16 Pooled Estimate of σ 2

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10-17 t-Based Confidence Interval for the Difference in Means (Variances Unknown)

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10-18 Example 10.3: The Catalyst Comparison Case

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10-19 t-Based Test About the Difference in Means: Variances Equal

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10-20 Example 10.4: The Catalyst Comparison Case

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10-21 t-Based Confidence Intervals and Tests for Differences with Unequal Variances

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Paired Difference Experiments Before, drew random samples from two different populations Now, have two different processes (or methods) Draw one random sample of units and use those units to obtain the results of each process

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10-23 Paired Difference Experiments Continued For instance, use the same individuals for the results from one process vs. the results from the other process E.g., use the same individuals to compare “before” and “after” treatments Using the same individuals, eliminates any differences in the individuals themselves and just comparing the results from the two processes

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10-24 Paired Difference Experiments #3 Let µ d be the mean of population of paired differences µ d = µ 1 – µ 2, where µ 1 is the mean of population 1 and µ 2 is the mean of population 2 Let d ̄ and s d be the mean and standard deviation of a sample of paired differences that has been randomly selected from the population d ̄ is the mean of the differences between pairs of values from both samples

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10-25 t-Based Confidence Interval for Paired Differences in Means

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10-26 Paired Differences Testing Rules

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10-27 Example 10.6 and 10.7: The Repair Cost Comparison Case

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Comparing Two Population Proportions by Using Large, Independent Samples Select a random sample of size n 1 from a population, and let p ̂ 1 denote the proportion of units in this sample that fall into the category of interest Select a random sample of size n 2 from another population, and let p ̂ 2 denote the proportion of units in this sample that fall into the same category of interest Suppose that n 1 and n 2 are large enough n 1 · p 1 ≥ 5, n 1 · (1 - p 1 ) ≥ 5, n 2 · p 2 ≥ 5, and n 2 · (1 – p 2 ) ≥ 5

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10-29 Comparing Two Population Proportions Continued Then the population of all possible values of p ̂ 1 - p̂ 2 Has approximately a normal distribution if each of the sample sizes n 1 and n 2 is large Has mean µ p ̂1 - p̂2 = p 1 – p 2 Has standard deviation

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10-30 Difference of Two Population Proportions

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10-31 Example 10.9 and 10.10: The Advertising Media Case

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Comparing Two Population Variances Using Independent Samples Population 1 has variance σ 1 2 and population 2 has variance σ 2 2 The null hypothesis H 0 is that the variances are the same H 0 : σ 1 2 = σ 2 2 The alternative is that one is smaller than the other That population has less variable measurements Suppose σ 1 2 > σ 2 2 More usual to normalize Test H 0 : σ 1 2 /σ 2 2 = 1 vs. σ 1 2 /σ 2 2 > 1

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10-33 Comparing Two Population Variances Using Independent Samples Continued Reject H 0 in favor of H a if s 1 2 /s 2 2 is significantly greater than 1 s 1 2 is the variance of a random of size n 1 from a population with variance σ 1 2 s 2 2 is the variance of a random of size n 2 from a population with variance σ 2 2 To decide how large s 1 2 /s 2 2 must be to reject H 0, describe the sampling distribution of s 1 2 /s 2 2 The sampling distribution of s 1 2 /s 2 2 is the F distribution

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10-34 F Distribution Figure 10.13

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10-35 F Distribution The F point F is the point on the horizontal axis under the curve of the F distribution that gives a right-hand tail area equal to The value of F depends on a (the size of the right-hand tail area) and df 1 and df 2 Different F tables for different values of Tables A.5 for = 0.10 Tables A.6 for = 0.05 Tables A.7 for = Tables A.8 for = 0.01

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10-36 Example 10.11: The Catalyst Comparison Case

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