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

2. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 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

3. 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

4. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 4 9.1: Identifying the Target Parameter µ1 - µ2 Mean difference; difference in averages Quantitative Data p1 - p2 Difference between proportions, percentages, fractions or rates; compare proportions Qualitative Data  1 2/  22 Ratio of variances; difference in variability or spread; compare variation Quantitative Data

5. 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 → µ 1 - µ 2 To construct a confidence interval or conduct a hypothesis test, we need the standard deviation: Singe sample Two samples

6. 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 ) 1. The mean of the sampling distribution is ( µ 1 -µ 2 ). 2. If the two samples are independent, the standard deviation of the sampling distribution (the standard error) is 3. The sampling distribution for (  1 -  2 ) is approximately normal for large samples.

7. 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 )

8. 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 (µ 1 - µ 2 )

9. 9.2: Comparing Two Population Means: Independent Sampling Private Colleges n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance:91.17 Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance:97.64 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 9 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. What does a 95% confidence interval tell us about retention rates? Source: National Center for Education Statistics

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

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

12. 9.2: Comparing Two Population Means: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 12 H 0 : (µ 1 - µ 2 ) = D 0 H a : (µ 1 - µ 2 ) > D 0 (< D 0 ) Rejection region: z z  ) Test Statistic: where H 0 : (µ 1 - µ 2 ) = D 0 H a : (µ 1 - µ 2 ) ≠ D 0 Rejection region: |z| > z  /2 Two-Tailed TestOne-Tailed Test

13. 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 (µ 1 - µ 2 ) 1. The two samples are randomly and independently selected from the target populations. 2. The sample sizes are both ≥ 30.

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

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

16. 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, s p 2

17. 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 (µ 1 - µ 2 ) The value of t is based on (n 1 + n 2 -2) degrees of freedom.

18. 9.2: Comparing Two Population Means: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 18 H 0 : (µ 1 - µ 2 ) = D 0 H a : (µ 1 - µ 2 ) > D 0 (< D 0 ) Rejection region: t t  ) Test Statistic: H 0 : (µ 1 - µ 2 ) = D 0 * H a : (µ 1 - µ 2 ) ≠ D 0 Rejection region: |t| > t  /2 Two-Tailed TestOne-Tailed Test

19. 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 (µ 1 - µ 2 ) 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.

20. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 20 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 H 0 : µ 1 = µ 2 9.2: Comparing Two Population Means: Independent Sampling

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

22. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 22 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 9.2: Comparing Two Population Means: Independent Sampling

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

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

25. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 25 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. 9.2: Comparing Two Population Means: Independent Sampling

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

27. 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?

28. 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.

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

30. 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.

31. 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

32. 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.

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

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

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

36. 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: MenWomen Yes290200 No50 n340250

37. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 37 9.4: Comparing Two Population Proportions: Independent Sampling MenWomen p.85.80 q.15.20 n340250 Calculate a 95% confidence interval on the difference in the opinions of men and women.

38. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 38 9.4: Comparing Two Population Proportions: Independent Sampling MenWomen p.85.80 q.15.20 n340250 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.

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

40. 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? 19921998 Survey n21,48423,199 Number who gambled4.6845,313 Proportion who gambled.218.229

41. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 41 9.4: Comparing Two Population Proportions: Independent Sampling 19921998 Survey n21,48423,199 Number who gambled 4.6845,313 Proportion who gambled.218.229

42. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 42 9.4: Comparing Two Population Proportions: Independent Sampling 19921998 Survey n21,48423,199 Number who gambled 4.6845,313 Proportion who gambled.218.229 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.

43. 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:

44. 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, n 1 = n 2

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

46. 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 n 1 and n 2 be?

47. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 47 9.5: Determining the Sample Size SE = 2.2  = 5% level.  1 2 =  2 2 =15. n 1 and n 2 = ?

48. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 48 9.5: Determining the Sample Size Sample size needed to estimate (p 1 - p 2 )  Given (1 -  ) and the sampling error (SE) required  Estimates of p 1 and p 2 will be needed The most conservative values are p 1 = p 2 =.5

49. 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 p 1 and p 2. What size do n 1 and n 2 need to be?

50. 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 p 1 and p 2. What size do n 1 and n 2 need to be?

51. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 51 9.6: Comparing Two Population Variances: Independent Sampling Two samples taken from the same population … should have means that are close to one another and … variances that are close to one another as well.

52. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 52 9.6: Comparing Two Population Variances: Independent Sampling Since variances do not follow a normal distribution (due to the squaring involved) we use a different technique for inferences about variances.

53. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 53 9.6: Comparing Two Population Variances: Independent Sampling

54. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 54 9.6: Comparing Two Population Variances: Independent Sampling If the ratio of variances gets too far from 1 in value, in either direction the likelihood that both samples share a population variance drops. For convenience the larger sample variance is usually put in the numerator.

55. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 55 9.6: Comparing Two Population Variances: Independent Sampling

56. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 56 9.6: Comparing Two Population Variances: Independent Sampling Home Runs, 2003 American League National League µ178.5169.25 22 1415.8951.4 n1416 Has the designated hitter produced a significant difference in homeruns in the two major leagues? To use the two-sample t-test, we should check the variance ratio to see if the assumption of equal variances is acceptable.

57. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 57 9.6: Comparing Two Population Variances: Independent Sampling Home Runs, 2003 American League National League µ178.5169.25 22 1415.8951.4 n1416

58. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 58 9.6: Comparing Two Population Variances: Independent Sampling Home Runs, 2003 American League National League µ178.5169.25 22 1415.8951.4 n1416 Since 1 is in the confidence interval, the assumption of equal variances needed for the t-test is not rejected.

59. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 59 9.6: Comparing Two Population Variances: Independent Sampling

60. In 2006, ten fast-growing economies had an average GDP growth rate of 8.69%, with a standard deviation of 1.7. Ten slow-growing economies had an average GDP growth rate of 2.29%, with a standard deviation of.58.* Do slow- and fast-growing economies have similar variability (  =5%)? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 60 * Source: euroekonom.com

61. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 61 9.6: Comparing Two Population Variances: Independent Sampling µ fast-growing = 8.69% µ slow-growing = 2.29%  2 fast-growing = 2.89  2 slow-growing =.34

62. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples 62 9.6: Comparing Two Population Variances: Independent Sampling µ fast-growing = 8.69% µ slow-growing = 2.29%  2 fast-growing = 2.89  2 slow-growing =.34 Reject the null hypothesis at the  =.05 level.