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Statistics

This chapter covers the concepts of constructing confidence intervals and conducting hypothesis tests to compare two populations based on sample data. Topics include comparing population means, proportions, and variances, as well as selecting sample sizes.

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Statistics

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  1. Statistics Chapter 9: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

  2. Where We’ve Been • Made inferences based on confidence intervals and tests of hypotheses • Studied confidence intervals and hypothesis tests for µ, p and 2 • Selected the necessary sample size for a given margin of error McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  3. Where We’re Going • Learn to use confidence intervals and hypothesis tests to compare two populations • Learn how to use these tools to compare two population means, proportions and variances • Select the necessary sample size for a given margin of error when comparing parameters from two populations McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  4. 9.1: Identifying the Target Parameter McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  5. 9.2: Comparing Two Population Means: Independent Sampling Point Estimators  → µ 1 - 2→ µ1 - µ2 To construct a confidence interval or conduct a hypothesis test, we need the standard deviation: Singe sample Two samples McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  6. 9.2: Comparing Two Population Means: Independent Sampling The Sampling Distribution for (1 - 2) • The mean of the sampling distribution is (µ1-µ2). • If the two samples are independent, the standard deviation of the sampling distribution (the standard error) is • The sampling distribution for (1 - 2) is approximately normal for large samples. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  7. 9.2: Comparing Two Population Means: Independent Sampling The Sampling Distribution for (1 - 2) McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  8. 9.2: Comparing Two Population Means: Independent Sampling Large Sample Confidence Interval for (µ1 - µ2 ) McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  9. 9.2: Comparing Two Population Means: Independent Sampling Private Colleges Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 Two samples concerning retention rates for first-year students at private and public institutions were obtained from the Department of Education’s data base to see if there was a significant difference in the two types of colleges. • n: 71 • Mean: 78.17 • Standard Deviation: 9.55 • Variance: 91.17 What does a 95% confidence interval tell us about retention rates? Source: National Center for Education Statistics McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  10. 9.2: Comparing Two Population Means: Independent Sampling Private Colleges Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 • n: 71 • Mean: 78.17 • Standard Deviation: 9.55 • Variance: 91.17 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  11. 9.2: Comparing Two Population Means: Independent Sampling Private Colleges Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 • n: 71 • Mean: 78.17 • Standard Deviation: 9.55 • Variance: 91.17 Since 0 is not in the confidence interval, the difference in the sample means appears to indicate a real difference in retention. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  12. 9.2: Comparing Two Population Means: Independent Sampling One-Tailed Test H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) ≠ D0 Rejection region: |z| >z/2 Two-Tailed Test H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) > D0 (< D0) Rejection region: z < -z(> z) Test Statistic: where McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  13. 9.2: Comparing Two Population Means: Independent Sampling Conditions Required for Valid Large-Sample Inferences about (µ1 - µ2) 1. The two samples are randomly and independently selected from the target populations. 2. The sample sizes are both ≥ 30. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  14. 9.2: Comparing Two Population Means: Independent Sampling Private Colleges Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 Let’s go back to the retention data and test the hypothesis that there is no significant difference in retention at privates and publics. • n: 71 • Mean: 78.17 • Standard Deviation: 9.55 • Variance: 91.17 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  15. 9.2: Comparing Two Population Means: Independent Sampling Test statistic: Reject the null hypothesis: McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  16. 9.2: Comparing Two Population Means: Independent Sampling For small samples, the t-distribution can be used with a pooled sample estimator of 2, sp2 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  17. 9.2: Comparing Two Population Means: Independent Sampling Small Sample Confidence Interval for (µ1 - µ2 ) The value of t is based on (n1 + n2 -2) degrees of freedom. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  18. 9.2: Comparing Two Population Means: Independent Sampling One-Tailed Test H0: (µ1 - µ2) = D0* Ha: (µ1 - µ2) ≠ D0 Rejection region: |t| >t/2 Two-Tailed Test H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) > D0 (< D0) Rejection region: t < -t(> t) Test Statistic: McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  19. 9.2: Comparing Two Population Means: Independent Sampling Conditions Required for Valid Small-Sample Inferences about (µ1 - µ2) 1. The two samples are randomly and independently selected from the target populations. 2. Both sampled populations have distributions that are approximately normal. 3. The population variances are equal. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  20. 9.2: Comparing Two Population Means: Independent Sampling • Does class time affect performance? • The test performance of students in two sections of international trade, meeting at different times, were compared. 8:00 a.m. Class Mean: 78 Standard Deviation: 14 Variance: 196 n: 21 9:30 a.m. Class Mean: 82 Standard Deviation: 17 Variance: 289 n: 21 With  = .05, test H0 : µ1 = µ2 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  21. 9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 78 Variance: 196 n: 21 9:30 a.m. Class Mean: 82 Variance: 289 n: 21 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  22. 9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 78 Variance: 196 n: 21 9:30 a.m. Class Mean: 82 Variance: 289 n: 21 With df = 18 + 24 – 2 = 40, t/2 = t.025 = 2.021. Since out test statistic t = -.812. |t| < t.025. Do not reject the null hypothesis McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  23. 9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 72 Variance: 154 n: 13 9:30 a.m. Class Mean: 86 Variance: 163 n: 21 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  24. 9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 72 Variance: 154 n: 13 9:30 a.m. Class Mean: 86 Variance: 163 n: 21 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  25. 9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 72 Variance: 154 n: 13 9:30 a.m. Class Mean: 86 Variance: 163 n: 21 Since |t| > t.025,df=26,, reject the null hypothesis. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  26. 9.3: Comparing Two Population Means: Paired Difference Experiments McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  27. 9.3: Comparing Two Population Means: Paired Difference Experiments Suppose ten pairs of puppies were housetrained using two different methods: one puppy from each pair was paper-trained, with the paper gradually moved outside, and the other was taken out every three hours and twenty minutes after each meal. The number of days until the puppies were considered housetrained (three days straight without an accident) were compared. Nine of the ten paper-trained dogs took longer than the other paired dog to complete training, with the average difference equal to 4 days, with a standard deviation of 3 days. What is a 90% confidence interval on the difference in successful training? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  28. 9.3: Comparing Two Population Means: Paired Difference Experiments Since 0 is not in the interval, one program does seem to work more effectively. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  29. 9.3: Comparing Two Population Means: Paired Difference Experiments McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  30. 9.3: Comparing Two Population Means: Paired Difference Experiments • Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” • Mean difference: $1.75 • Standard Deviation: $10.35 • Test at the 95% level that the difference in the two stores is zero. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  31. 9.3: Comparing Two Population Means: Paired Difference Experiments Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” • Mean difference: $1.75 • Standard Deviation: $10.35 •  = .05 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  32. 9.3: Comparing Two Population Means: Paired Difference Experiments Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” • Mean difference: $1.75 • Standard Deviation: $10.35 •  = .05 The critical value of z.05 is 1.96, so we would reject this null hypothesis. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  33. 9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  34. 9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  35. 9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  36. 9.4: Comparing Two Population Proportions: Independent Sampling • A group of men and women were asked their opinions on the following important issue: Are the Three Stooges funny? The results are as follow: McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  37. 9.4: Comparing Two Population Proportions: Independent Sampling • Calculate a 95% confidence interval on the difference in the opinions of men and women. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  38. 9.4: Comparing Two Population Proportions: Independent Sampling • Calculate a 95% confidence interval on the difference in the opinions of men and women. Since 0 is in the confidence interval, we cannot rule out the possibility that both genders find the Stooges equally funny. Nyuk nyuk nyuk. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  39. 9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  40. 9.4: Comparing Two Population Proportions: Independent Sampling • Randy Stinchfield of the University of Minnesota studied the gambling activities of public school students in 1992 and 1998 (Journal of Gambling Studies, Winter 2001). His results are reported below: • Do these results represent a statistically significant difference at the  = .01 level? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  41. 9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  42. 9.4: Comparing Two Population Proportions: Independent Sampling Since the computed value of z, -2.786, is of greater magnitude than the critical value, 2.576, we can reject the null hypothesis at the  = .01 level. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  43. 9.4: Comparing Two Population Proportions: Independent Sampling • For valid inferences • The two samples must be independent • The two sample sizes must be large: McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  44. 9.5: Determining the Sample Size • With a given level of confidence, and a specified sampling error, it is possible to calculate the required sample size • Typically, n1 = n2 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  45. 9.5: Determining the Sample Size • Sample size needed to estimate (µ1 - µ1) • Given (1 -  ) and the sampling error (SE) required • Estimates of 1 and 2 will be needed McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  46. 9.5: Determining the Sample Size • Suppose you need to estimate the difference between two population means to within 2.2 at the  = 5% level. You have good reason to believe the two variances are equal to each other, and equal 15. • How large must n1 and n2 be? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  47. 9.5: Determining the Sample Size • SE = 2.2 •  = 5% level. • 12 = 22 =15. • n1 and n2 = ? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  48. 9.5: Determining the Sample Size • Sample size needed to estimate (p1 - p2) • Given (1 -  ) and the sampling error (SE) required • Estimates of p1 and p2 will be needed • The most conservative values are p1 = p2 =.5 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  49. 9.5: Determining the Sample Size • Suppose you need to calculate a 90% confidence interval of width .05, with no information about possible values of p1 and p2. • What size do n1 and n2 need to be? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

  50. 9.5: Determining the Sample Size • Suppose you need to calculate a 90% confidence interval of width .05, with no information about possible values of p1 and p2. • What size do n1 and n2 need to be? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

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