Two-Sample Hypothesis Tests

2. Two-Sample Hypothesis Tests Two-Sample Tests Comparing Two Proportions Comparing Two Means: Independent Samples Comparing Two Means: Paired Samples Comparing Two Variances

3. Two-Sample Tests A Two-sample test compares two sample estimates with each other. A one-sample test compares a sample estimate against a non-sample benchmark.

4. Two-Sample Tests If the two sample statistics differ by more than the amount attributable to chance, then we conclude that the samples came from populations with different parameter values.

5. Two-Sample Tests State the hypotheses Set up the decision rule Insert the sample statistics Make a decision based on the critical values or using p-values If our decision is wrong, we could commit a type I or type II error. Larger samples are needed to reduce type I or type II errors.

6. Comparing Two Proportions To compare two population proportions, p1, p2, use the following hypotheses

7. Comparing Two Proportions The sample proportion p1 is a point estimate of p1:

8. Comparing Two Proportions If H0 is true, there is no difference between p1 and p2, so the samples are pooled (averaged) into one “big” sample to estimate the common population proportion.

9. Comparing Two Proportions If the samples are large, p1 – p2 may be assumed normally distributed. The test statistic is the difference of the sample proportions divided by the standard error of the difference. The standard error is calculated by using the pooled proportion. The test statistic for the hypothesis p1 = p2 is:

10. Comparing Two Proportions The test statistic for the hypothesis p1 = p2 may also be written as:

11. Comparing Two Proportions Step 1: State the hypotheses Step 2: State the decision ruleChoose a (the level of significance) and determine the critical value(s). Step 3: Calculate the Test StatisticAssuming that p1 = p2, use a pooled estimate of the common proportion.

12. Comparing Two Proportions Step 4: Make the decisionReject H0 if the test statistic falls in the rejection region(s) as defined by the critical value(s).

13. Comparing Two Proportions The p-value is the level of significance that allows us to reject H0. The p-value indicates the probability that a sample result would occur by chance if H0 were true. The p-value can be obtained from Appendix C-2 or Excel using =NORMDIST(z). A smaller p-value indicates a more significant difference.

14. Comparing Two Proportions Check the normality assumption withnp > 10 and n(1-p) > 10. Each of the two samples must be checked separately using each sample proportion in place of p. If either sample proportion is not normal, their difference cannot safely be assumed normal. The sample size rule of thumb is equivalent to requiring that each sample contains at least 10 “successes” and at least 10 “failures.”

15. Comparing Two Proportions If sample sizes do not justify the normality assumption, treat each sample as a binomial experiment. If the samples are small, the test is likely to have low power.

16. Comparing Two Proportions MegaStat gives the option of entering sample proportions or the fractions. MINITAB gives the option of nonpooled proportions.

17. Comparing Two Proportions The confidence interval for p1 – p2 without pooling the samples is:

18. Comparing Two Proportions Testing for equality is a special case of testing for a specified difference D0 between two proportions.

19. Comparing Two Means: Independent Samples The hypotheses for comparing two independent population means m1 and m2 are:

20. Comparing Two Means: Independent Samples If the population variances s12 and s22 are known, then use the normal distribution. If population variances are unknown and estimated using s12 and s22, then use the Students t distribution.

21. Comparing Two Means: Independent Samples Excel’s Tools | Data Analysis menu handles all three cases.

22. Comparing Two Means: Independent Samples When the variances are known, use the normal distribution for the test (assuming a normal population). The test statistic is:

23. Comparing Two Means: Independent Samples Since the variances are unknown, they must be estimated and the Student’s t distribution used to test the means. Assuming the population variances are equal, s12 and s22 can be used to estimate a common pooled variance sp2.

24. Comparing Two Means: Independent Samples The test statistic is

25. Comparing Two Means: Independent Samples If the unknown variances are assumed to be unequal, they are not pooled together.

26. Comparing Two Means: Independent Samples

27. Comparing Two Means: Independent Samples Step 1: State the hypotheses Step 2: State the decision ruleChoose a (the level of significance) and determine the critical value(s). Step 3: Calculate the Test Statistic

28. Comparing Two Means: Independent Samples Step 4: Make the decisionReject H0 if the test statistic falls in the rejection region(s) as defined by the critical value(s).

29. Comparing Two Means: Independent Samples If the sample sizes are equal, the Case 2 and Case 3 test statistics will be identical, although the degrees of freedom may differ. If the variances are similar, the two tests will usually agree. If no information about the population variances is available, then the best choice is Case 3. The fewer assumptions, the better.

30. Comparing Two Means: Independent Samples

31. Comparing Two Means: Independent Samples

32. Comparing Two Means: Independent Samples

33. Comparing Two Means: Paired Samples

34. Comparing Two Means: Paired Samples

35. Comparing Two Means: Paired Samples

36. Comparing Two Means: Paired Samples

37. Comparing Two Means: Paired Samples Step 3: Calculate the test statistic t Step 4: Make the decisionReject H0 if the test statistic falls in the rejection region(s) as defined by the critical values.

38. Comparing Two Means: Paired Samples Excel gives you the option of choosing either a one-tailed or two-tailed test and also gives the p-value.

39. Comparing Two Means: Paired Samples A two-tailed test for a zero difference is equivalent to asking whether the confidence interval for the true mean difference md includes zero.

40. Comparing Two Variances To test whether two population means are equal, we may also need to test whether two population variances are equal. The hypotheses may be stated as

41. Comparing Two Variances An equivalent way to state these hypotheses would be to use ratios since the variance can never be less than zero and it would not make sense to take the difference between two variances.

42. Comparing Two Variances The test statistic is the ratio of the sample variances:

43. Comparing Two Variances If the test statistic is far below 1 or above 1, we would reject the hypothesis of equal population variances. The numerator s12 has degrees of freedom n1 = n1 – 1 and the denominator s22 has degrees of freedom n2 = n2 – 1. The F distribution is skewed with the mean > 1 and its mode < 1.

44. Comparing Two Variances Critical values for the F test are denoted FL (left tail) and FR (right tail). A right-tail critical value FR may be found from Appendix F using n1 and n2 degrees of freedom. FR = Fn1,n2 A left-tail critical value FR may be found by reversing the numerator and denominator degrees of freedom, finding the critical value from Appendix F and taking its reciprocal: FL = 1/Fn2,n1

45. Comparing Two Variances

46. Comparing Two Variances Step 3: Calculate the test statistic Step 4: Make the decisionReject H0 if the test statistic falls in the rejection regions as defined by the critical values.

47. Comparing Two Variances

48. Comparing Two Variances

49. Comparing Two Variances

50. Comparing Two Variances