solpracmidS09 - PSTAT 120C Solutions to Practice Midterm...

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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 one-sided alternative, the P -value 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 two-sided test and 11 for a one-sided test. Thus, we get a significant difference between the two samples in either the one-sided or two-side 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 ( 0 . 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
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(b) The t statistic is t = ¯ d p s 2 /n = - 0 . 45 p 0 . 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 P -value of 0.125. (d) The signed-rank 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 for α = 0 . 1 is 4. Therefore, the signed-rank test shows a significant difference between the prices in the two samples.
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