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Unformatted text preview: PSTAT 120C: Solutions to Practice Midterm Questions April 29, 2009 1. 15.4 (a) There are 7 out of 10 of the twins where student A did better than student B. The probability of this occurring is P { M ≥ 7 } = 2 10 10 7 + 10 8 + 10 9 + 10 10 = 2 10 (120 + 45 + 10 + 1) = 176 1024 so that the P value is 2 176 1024 = 0 . 34375 . This is not significant at the α = 0 . 05 because the P value is bigger than α . (b) For the onesided alternative, the Pvalue is P = 176 1024 = 0 . 171875 15.13 The ranks of the differences are Pair Difference Rank Sign Pair Difference Rank Sign 1 28 10 + 6 2 1 2 5 4 + 7 7 5 + 3 4 3 8 9 7 + 4 15 9 + 9 3 2 5 8 6 + 10 13 8 + The sum of the ranks of the negative observations is T = 3 + 1 + 2 = 6 . The critical value for α = 0 . 05 from Table 9 is 8 for a twosided test and 11 for a onesided test. Thus, we get a significant difference between the two samples in either the onesided or twoside tests. Therefore, the test is more significant than the sign test. This is often true because we are able to use more information in the data (from the ranks) to get a more powerful test. 2. (a) The estimate of the variance of the difference is s 2 = 7 6 ( . 44464 ( . 45) 2 ) = 0 . 2825 The critical values for a 90% confidence interval when there are 6 degrees of freedom are χ 2 6 ,. 95 = 1 . 63539 and χ 2 6 ,. 05 = 12 . 5916. The confidence interval is therefore " ( n 1) s 2 χ 2 6 ,. 05 , ( n 1) s 2 χ 2 6 ,. 95 # 6(0 . 2825) 12 . 5916 , 6(0 . 2825) 1 . 63539 [0 . 135 , 1 . 04] 1 (b) The t statistic is t = ¯ d p s 2 /n = . 45 p . 2825 / 7 = 2 . 24 . The critical value with 6 degrees of freedom is 1.943, and therefore there is a significant difference between the prices. (c) The sign test notes that M = 1 of the stores has sugar more expensive than eggs. The probability of this happening by chance is P { M ≤ 1 } = P { M = 0 } + P { M = 1 } = 2 7 (1 + 7) = . 0625 giving a Pvalue of 0.125. (d) The signedrank test statistic first ranks the seven stores by the magnitudes of the differ ence between the observations. Store Sugar, x Eggs, y  x y  Rank 1 $1.40 $2.00 $0.60 4 2 $1.55 $1.70 $0.15 2 3 $1.85 $1.45 $0.40 3 4 $0.95 $1.00 $0.05 1 5 $0.85 $1.90 $1.05 7 6 $1.00 $1.75 $0.75 5 7 $1.15 $2.10 $0.95 6 So that the rank sum is 3. From Table 9 in the text we get the critical value forSo that the rank sum is 3....
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This note was uploaded on 10/02/2009 for the course PSTAT 5E taught by Professor Eduardomontoya during the Spring '08 term at UCSB.
 Spring '08
 EduardoMontoya

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