TwoSample Hypothesis Tests Copyright c 2008 by The
Two-Sample Hypothesis Tests Copyright (c) 2008 by The Mc. Graw-Hill Companies. This material is intended solely for educational purposes by licensed users of Learning. Stats. It may not be copied or resold for profit.
Two-Sample Tests Parameter Left-Tail Test Two-Tail Test Right-Tail Test 2 Means H 0 : m 1 m 2 H 1 : m 1 < m 2 H 0 : m 1 = m 2 H 1 : m 1 m 2 H 0 : m 1 m 2 H 1 : m 1 > m 2 2 Proportions H 0 : p 1 p 2 H 1 : p 1 < p 2 H 0 : p 1 = p 2 H 1 : p 1 p 2 H 0 : p 1 p 2 H 1 : p 1 > p 2 2 Variances H 0 : s 1 2 s 2 2 H 1 : s 1 2 < s 2 2 H 0 : s 1 2 = s 2 2 H 1 : s 1 2 s 2 2 H 0 : s 1 2 s 2 2 H 1 : s 1 2 > s 2 2
Example: Two Means Given: Resting systolic blood pressure for 12 men and 17 women over age 65 on the same medication. The men show a mean of 137. 50 with a standard deviation of 7. 2926, while the women show a mean of 135. 29 with a standard deviation of 7. 5396. Question: Do the men have higher blood pressure than the women? Answer: No. Assuming equal variances, the large p-value (p = 0. 219) in a right-tail test indicates no significant difference at a = 0. 05. If we assume unequal variances the conclusion is the same (p = 0. 218).
Data: Two Means
Hypotheses: Two Means H 0: m 1 m 2 H 1: m 1 > m 2 Note Since the population standard deviations are unknown, we use a t-test. Assuming equal (pooled) variances, we would use d. f. = n 1 + n 2 2 = 12 + 17 – 2 = 27. Assuming unequal variances, using Welch’s correction, the reduced d. f. = 24. The formula for Welch’s correction is too complex to show here.
Excel: Two Means Assuming Equal Population Variances Assuming Unequal Population Variances
Excel: Two Means Results from Excel’s Tools > Data Analysis t-Test: Two-Sample Assuming Equal Variances Male Mean t-Test: Two-Sample Assuming Unequal Variances Female Male 137. 5 135. 2941176 Variance 53. 181818 56. 84558824 Std Dev 7. 2925865 7. 539601331 12 17 Observations Pooled Variance Hypothesized Mean Diff df 55. 352941 0 27 t Stat 0. 7863721 P(T<=t) one-tail 0. 2192486 t Critical one-tail 1. 703288 P(T<=t) two-tail 0. 4384973 t Critical two-tail 2. 0518291 Mean Variance Observations Hypothesized Mean Diff df Female 137. 5 135. 2941176 53. 1818 56. 84558824 12 17 0 24 t Stat 0. 791066985 P(T<=t) one-tail 0. 218326615 t Critical one-tail 1. 710882316 P(T<=t) two-tail 0. 436653229 t Critical two-tail 2. 063898137 Note Because the sample variances are similar, either method gives about the same result.
Calculations: Formulas If we assume equal variances: If we assume unequal variances: Note The results are similar in this example
Two Means: Minitab
Two Means: Minitab Note You must specify the variance assumption. Copyright Notice Portions of MINITAB Statistical Software input and output contained in this document are printed with permission of Minitab, Inc. MINITAB TM is a trademark of Minitab Inc. in the United States and other countries and is used herein with the owner's permission.
Minitab: Two Means Because the sample variances are similar in this example, either method gives about the same result.
Example: Two Proportions Given: Post-stroke 1 -year death rates for patients taking a new drug compared with a control group. Question: Does the new significantly reduce death rates at a = 0. 05? Answer: No. For a left-tail test, the p-value (p = 0. 0779) exceeds a = 0. 05. However, the result would be significant at a = 0. 10, suggesting the need for further study with a larger sample size for more power.
Data: Two Proportions =NORMSDIST(-1. 419)
Hypotheses: Two Proportions H 0: p 1 p 2 H 1: p 1 < p 2 Note To ensure normality, this test requires large samples. A good rule* is that n 1 pc 10 and n 1(1 -pc) 10 and n 2 pc 10 and n 2(1 -pc) 10 where pc is the combined (pooled) sample proportion. *some textbooks allow 5 instead of 10
Calculations: Formulas General formula: Pooled proportion: Calculations for this sample: Note n 1 pc = (352)(0. 032927) = 11. 6 and n 1(1 -pc) = 352(0. 967073) = 340. 4 and n 2 pc =(468)(0. 032927) = 15. 4 and n 2(1 -pc) = (468)(0. 967073) = 145. 6. Since these all exceed 10, it is safe to assume normality.
Two Proportions: Minitab Copyright Notice Portions of MINITAB Statistical Software input and output contained in this document are printed with permission of Minitab, Inc. MINITAB TM is a trademark of Minitab Inc. in the United States and other countries and is used herein with the owner's permission.
Two Proportions: Minitab Copyright Notice Portions of MINITAB Statistical Software input and output contained in this document are printed with permission of Minitab, Inc. MINITABTM is a trademark of Minitab Inc. in the United States and other countries and is used herein with the owner's permission.
Minitab: Two Proportions
Example: Two Variances Given: Resting systolic blood pressure for 12 men and 17 women over age 65 on the same medication. The men show a mean of 137. 50 with a standard deviation of 7. 2926, while the women show a mean of 135. 29 with a standard deviation of 7. 5396. Question: Do the men and women have the same variance in blood pressure? Answer: Yes. The large p-value (p = 0. 934) in Minitab's two-tail test indicates no significant difference at a = 0. 05. We ignore the means since they are not relevant to this test.
Data: Two Variances
Hypotheses: Two Variances H 0: s 12 = s 22 H 1: s 12 s 22 Note This is a two-tail test. It will show whether or not variances are the same. However, it could also be a one-tail test in either direction if we wished.
Excel: Two Variances
Excel: Two Variances Results from Excel’s Tools > Data Analysis Note F is the ratio of the sample variances
Calculations: Formulas General formula: Note If the population variances are the same, the ratio of the sample variances should be near 1. For this sample:
Two Variances: Minitab
Two Variances: Minitab Copyright Notice Portions of MINITAB Statistical Software input and output contained in this document are printed with permission of Minitab, Inc. MINITAB TM is a trademark of Minitab Inc. in the United States and other countries and is used herein with the owner's permission.
Two Variances: Minitab Note Minitab shows confidence intervals for the separate population standard deviations. Since they overlap, we can see that there is no significant difference in a two-tailed test.
Two Variances: Minitab Note Minitab gives several tests (not just an F-test). Their p-values are in general agreement, indicating no significant difference in variances.
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