Rejection region outside 0 2 17 104 25 4 426 then the

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rejection region: outside 0 ± 2 . 17 * 104 / 25 = ± 4 . 426. Then the power is P { Z < - 9 . 426 / 2 . 04 } + P { Z > - 0 . 574 / 2 . 04 } = 0 + P { Z > - 0 . 281 } = 1 - 0 . 3897 = 0 . 61. 4(a) ˆ p A = 63 / 220 = . 2864, ˆ p C = 59 / 80 = . 7375 and ˆ p = (63 + 59) / 300 = . 4067. We can approximate the binomial with a normal distribution because 220 * ˆ p = 41 . 3 > 5 and 220 * ˆ q = 178 . 7 > 5 and similarly 80 * ˆ p = 37 . 4 > 5 and 80 * ˆ q > 5. Then z = ( . 7375 - . 2864) / . 4067 * . 5933 * (1 / 220 + 1 / 80) = 7 . 034. Us- ing the normal distribution we get p < 2 * 2 . 9 * 10 - 7 is way smaller than . 0001 and way smaller than α = . 10. We strongly reject p A = p C : there is strong evidence that the locus is linked to some genetic region affect- ing flowering time, the C allele being linked to early flowering. Assumptions include random sampling of and independence among plants. (b) Observations are clearly paired: each leaf observa- tion is paired to the root observation made on the same plant. Therefore, one would use a two-sided paired-sample t-test, provided that the distribution of difference (leaf expression -root expression) is not too far from a normal distribution. Note: there is no need to assume that σ 1 = σ 2 and no need to assume normality of gene expression. Only the normality of the expression difference is needed. 5(a) false : we mostly use a t-distribution. The normal distribution is only used when the variance is known. true true because z-quantiles are smaller than t-quantiles. false : the confidence interval is centered at the sam- ple mean (¯ y , which is random) while the not-rejection region is centered at the null hypothesis μ 0 (not ran- dom - known before collecting the data) true true : they both have the same length, which is twice z α/ 2 * σ/ n . Summary of grades: Frequency 20 40 60 80 100 0 10 20 30 40 ●● 76 83 89 1
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