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Chapter 10

Chapter 10. Statistical Inferences Based on Two Samples. Chapter Outline. 10.1 Comparing Two Population Means by Using Independent Samples: Variances Known 10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown 10.3 Paired Difference Experiments

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Chapter 10

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  1. Chapter 10 Statistical Inferences Based onTwo Samples

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

  3. 10.1 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

  4. Sampling Distribution of theDifference of Two Sample Means #1 • Suppose population 1 has mean µ1 and variance σ12 • From population 1, a random sample of size n1 is selected which has mean x1 and variance s12 • Suppose population 2 has mean µ2 and variance σ22 • From population 2, a random sample of size n2 is selected which has mean x2 and variance s22 • Then the sample distribution of the difference of two sample means…

  5. Sampling Distribution of theDifference of Two Sample Means #2 • Is normal, if each of the sampled populations is normal • Approximately normal if the sample sizes n1 and n2 are large • Has mean µx1–x2 = µ1 – µ2 • Has standard deviation

  6. Sampling Distribution of theDifference of Two Sample Means #3 Figure 10.1

  7. z-Based Confidence Interval for the Difference in Means (Variances Known) • A 100(1 – ) percent confidence interval for the difference in populations µ1–µ2 is

  8. z-Based Test About the Difference in Means (Variances Known) • Test the null hypothesis aboutH0: µ1 – µ2 = D0 • D0 = µ1 – µ2 is the claimed difference between the population means • D0 is a number whose value varies depending on the situation • Often D0 = 0, and the null means that there is no difference between the population means

  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 n1 and n2 are large

  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 ...

  11. z-Based Test About the Difference in Means (Variances Known) • Reject H0: µ1 – µ2 = D0 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…

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

  13. Example 10.2: The Bank Customer Waiting Time Case

  14. 10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown • Generally, the true values of the population variances σ12 and σ22 are not known • They have to be estimated from the sample variances s12 and s22, respectively

  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: • If it can be assumed that σ12 = σ22 = σ2, then calculate the “pooled estimate” of σ2 • If σ12 ≠ σ22, then use approximate methods

  16. Pooled Estimate of σ2

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

  18. Example 10.3: The Catalyst Comparison Case

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

  20. Example 10.4: The Catalyst Comparison Case

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

  22. 10.3 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

  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

  24. Paired Difference Experiments #3 • Let µd be the mean of population of paired differences • µd = µ1 – µ2, where µ1 is the mean of population 1 and µ2 is the mean of population 2 • Let d̄ and sd be the mean and standard deviation of a sample of paired differences that has been randomly selected from the population • d̄ is the mean of the differences between pairs of values from both samples

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

  26. Paired Differences Testing Rules

  27. Example 10.6 and 10.7: The Repair Cost Comparison Case

  28. 10.4 Comparing Two Population Proportions by Using Large, Independent Samples • Select a random sample of size n1 from a population, and let p̂1 denote the proportion of units in this sample that fall into the category of interest • Select a random sample of size n2 from another population, and let p̂2 denote the proportion of units in this sample that fall into the same category of interest • Suppose that n1 and n2 are large enough • n1·p1 ≥ 5, n1·(1 - p1) ≥ 5, n2·p2 ≥ 5, and n2·(1 – p2) ≥ 5

  29. Comparing Two Population Proportions Continued • Then the population of all possible values of p̂1 - p̂2 • Has approximately a normal distribution if each of the sample sizes n1 and n2 is large • Has mean µp̂1 - p̂2 = p1 – p2 • Has standard deviation

  30. Difference of Two Population Proportions

  31. Example 10.9 and 10.10: The Advertising Media Case

  32. 10.5 Comparing Two Population Variances Using Independent Samples • Population 1 has variance σ12 and population 2 has variance σ22 • The null hypothesis H0 is that the variances are the same • H0: σ12 = σ22 • The alternative is that one is smaller than the other • That population has less variable measurements • Suppose σ12 > σ22 • More usual to normalize • Test H0: σ12/σ22 = 1 vs. σ12/σ22 > 1

  33. Comparing Two Population Variances Using Independent Samples Continued • Reject H0 in favor of Ha if s12/s22 is significantly greater than 1 • s12 is the variance of a random of size n1 from a population with variance σ12 • s22 is the variance of a random of size n2 from a population with variance σ22 • To decide how large s12/s22 must be to reject H0, describe the sampling distribution of s12/s22 • The sampling distribution of s12/s22 is the F distribution

  34. F Distribution Figure 10.13

  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 df1 and df2 • Different F tables for different values of  • Tables A.5 for  = 0.10 • Tables A.6 for  = 0.05 • Tables A.7 for  = 0.025 • Tables A.8 for  = 0.01

  36. Example 10.11: The Catalyst Comparison Case

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