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【正文】 pool the two p estimates Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 937 Two Population Proportions Population proportions ? ? ? ???????????????212121n1n1)p1(pppppzThe test statistic for p1 – p2 is: (continued) Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 938 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H0: p1 – p2 ? 0 HA: p1 – p2 0 Upper tail test: H0: p1 – p2 ≤ 0 HA: p1 – p2 0 Twotailed test: H0: p1 – p2 = 0 HA: p1 – p2 ≠ 0 ? ?/2 ?/2 ? z? z?/2 z? z?/2 Reject H0 if z z? Reject H0 if z z? Reject H0 if z z?/2 or z z?/2 Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 939 Example: Two population Proportions Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A? ? In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes ? Test at the .05 level of significance Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 940 ? The hypothesis test is: H0: p1 – p2 = 0 (the two proportions are equal) HA: p1 – p2 ≠ 0 (there is a significant difference between proportions) ? The sample proportions are: ? Men: p1 = 36/72 = .50 ? Women: p2 = 31/50 = .62 . 5 4 91 2 2675072 3136nn xxp2121 ????????? The pooled estimate for the overall proportion is: Example: Two population Proportions (continued) Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 941 The test statistic for p1 – p2 is: Example: Two population Proportions (continued) .025 .025 Decision: Do not reject H0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. ? ? ? ?? ? ? ?1 . 3 1501721. 5 4 9 )(1. 5 4 90. 6 2. 5 0n1n1p)(1pppppz212121???????????????????????????Reject H0 Reject H0 Critical Values = 177。 For ? = .05 Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 942 Two Sample Tests in EXCEL For independent samples: ? Independent sample Z test with variances known: ? Tools | data analysis | ztest: two sample for means ? Independent sample Z test with large sample ? Tools | data analysis | ztest: two sample for means ? If the population variances are unknown, use sample variances For paired samples (t test): ? Tools | data analysis… | t test: paired two sample for means Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 943 Two Sample Tests in PHStat Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 944 Two Sample Tests in PHStat Input Output Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 945 Two Sample Tests in PHStat Input Output Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 946 Chapter Summary ? Compared two independent samples ? Formed confidence intervals for the differences between two means ? Performed Z test for the differences in two means ? Performed t test for the differences in two means ? Compared two related samples (paired samples) ? Formed confidence intervals for the paired difference ? Performed paired sample t tests for the mean difference ? Compared two population proportions ? Formed confidence intervals for the difference between two population proportions ? Performed Ztest for two population proportions