Elementary Statistics Exam 3 Solutions for Math 220: Inference and Hypothesis Testing - Pr, Exams of Statistics

Download Elementary Statistics Exam 3 Solutions for Math 220: Inference and Hypothesis Testing - Pr and more Exams Statistics in PDF only on Docsity! Math 220 : Elementary Statistics Kane Nashimoto Exam 3 Solutions Spring 2006 1. Sampling distribution. Note : X̄ , mean unit price of Porterhouse steak, is normal with µX̄ = µ = 8.45 and σX̄ = σ/ √ n = 0.63/ √ 25. (a) P (X > 9.25) = P ( X − µ σ > 9.25 − 8.45 0.63 ) = P (Z > 1.27) = 1 − P (Z ≤ 1.27) = 1 − .8980 = .1020 (b) P (X̄ > 8.75) = P ( X̄ − µ σ/ √ n > 8.75 − 8.45 0.63/ √ 25 ) = P (Z > 2.38) = 1 − P (Z ≤ 2.38) = 1 − .9913 = .0087 2. Inference about proportion. (a) H0 : π = .75 vs. Ha : π < .75 p = 101/150 = .673 z∗ = p − π0 √ π0(1 − π0)/n = .673 − .75 √ (.75)(1 − .75)/150 = −2.178 Critical value : −1.645 (z.05 = 1.645) Reject H0. (π < .75) (b) p-value = P (Z ≤ −2.15) = .0158 3. Inference about mean with σ known. (a) n = ( z.05/2 · σ B ) 2 = ( 1.960 · 2.8 0.55 ) 2 = 99.56 ⇒ 100 (b) x̄ ± z.05/2 σ√ n ; 67.4 ± 1.960 2.8√ 75 ; 67.4 ± 0.634 ; (66.766 , 68.034) 4. Inference about mean with σ unknown. (a) H0 : µ = 1.28 vs. Ha : µ 6= 1.28 t∗ = x̄ − µ0 s/ √ n = 1.36 − 1.28 0.87/ √ 20 = 0.411 Critical values : ±2.093 (t.05/2, 20−1 = 2.093) Retain H0. (µ ≈ 1.28) (b) The 95% confidence interval should contain µ0 = 1.28. The result of (a) indicates that 1.28 is a plausible value of µ and, therefore, must be contained in the confidence interval. (c) x̄ ± t.05/2, 20−1 s√ n ; 1.36 ± 2.0930.87√ 20 ; 1.36 ± 0.407 ; (0.953 , 1.767) (Note that this interval does contain 1.28.)