【正文】 esis Testing for Paired Samples ? ?/2 ?/2 ? t? t?/2 t? t?/2 Reject H0 if t t? Reject H0 if t t? Reject H0 if t t?/2 or t t?/2 Where t has n 1 . (continued) Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 927 Calculating the Test Statistic ? ? ? ? ? ? ? ? 1 . 2 2 5 6225211 . 1 61251 . 3 01212nns1ns1ns 2221222211p ?????????????? ? ? ? ? ?2 . 0 4 02512111 . 2 2 5 602 . 5 33 . 2 7n1n1sμμxxz21p2121 ??????????The test statistic is: Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 924 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ? 30 σ1 and σ2 unknown, n1 or n2 30 σ1 and σ2 unknown, small samples Where t?/2 has (n1 + n2 – 2) ., and ? ? ? ?2nns1ns1ns21222211p ??????? ? ? ?21p2121n1n1sμμxxz?????The test statistic for μ1 – μ2 is: * Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 920 Hypothesis Tests for Two Population Proportions Lower tail test: H0: μ1 ? μ2 HA: μ1 μ2 ., H0: μ1 – μ2 ? 0 HA: μ1 – μ2 0 Upper tail test: H0: μ1 ≤ μ2 HA: μ1 μ2 ., H0: μ1 – μ2 ≤ 0 HA: μ1 – μ2 0 Twotailed test: H0: μ1 = μ2 HA: μ1 ≠ μ2 ., H0: μ1 – μ2 = 0 HA: μ1 – μ2 ≠ 0 Two Population Means, Independent Samples Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 916 Paired Samples Tests Means of 2 Related Populations ? Paired or matched samples ? Repeated measures (before/after) ? Use difference between paired values: ? Eliminates Variation Among Subjects ? Assumptions: ? Both Populations Are Normally Distributed ? Or, if Not Normal, use large samples Paired samples d = x1 x2 Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 912 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ? 30 σ1 and σ2 unknown, n1 or n2 30 σ1 and σ2 unknown, small samples Assumptions: ? populations are normally distributed ? the populations have equal variances ? samples are independent * Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 98 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ? 30 σ1 and σ2 unknown, n1 or n2 30 ? ?222121/221 nσnσzxx ??? ?The confidence interval for μ1 – μ2 is: σ1 and σ2 known (continued) * Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 94 Difference Between Two Means Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ? 30 σ1 and σ2 unknown, n1 or n2 30 Goal: Form a confidence interval for the difference between two population means, μ1 – μ2 The point estimate for the difference is x1 – x2 * Business Statistics: A DecisionMaking Approach, 6e 169。Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 93 Estimation for Two Populations Estimating two population values Population means, independent samples Paired samples Population proportions Group 1 vs. independent Group 2 Same group before vs. after treatment Proportion 1 vs. Proportion 2 Examples: Business Statistics: A DecisionMaking Approach, 6e 169。 2020 PrenticeHall, Inc. Chap 97 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ? 30 σ1 and σ2 unknown, n1 or n2 30 …and the standard error of x1 – x2 is When σ1