ch8 - Chapter 8 Tests of Statistical Hypotheses 8.1 Tests...

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Unformatted text preview: Chapter 8 Tests of Statistical Hypotheses 8.1 Tests about Proportions 8.1–2 (a) C = { x : x = 0 , 1 , 2 } ; (b) α = P ( X = 0 , 1 , 2; p = 0 . 6) = (0 . 4) 4 + 4(0 . 6)(0 . 4) 3 + 6(0 . 6) 2 (0 . 4) 2 = 0 . 5248; β = P ( X = 3 , 4; p = 0 . 4) = 4(0 . 4) 3 (0 . 6) + (0 . 4) 4 = 0 . 1792 . OR (a ) C = { x : x = 0 , 1 } ; (b ) α = P ( X = 0 , 1; p = 0 . 6) = (0 . 4) 4 + 4(0 . 6)(0 . 4) 3 = 0 . 1792; β = P ( X = 2 , 3 , 4; p = 0 . 4) = 6(0 . 4) 2 (0 . 6) 2 + 4(0 . 4) 3 (0 . 6) + (0 . 4) 4 = 0 . 5248 . 8.1–4 Using Table II in the Appendix, (a) α = P ( Y ≥ 13; p = 0 . 40) = 1- . 8462 = 0 . 1538; (b) β = P ( Y ≤ 12; p = 0 . 60) = P (25- Y ≥ 25- 12) where 25- Y is b (25 , . 40) = 1- . 8462 = 0 . 1538 . 8.1–6 (a) z = y/n- 1 / 6 p (1 / 6)(5 / 6) /n ≤ - 1 . 645; (b) z = 1265 / 8000- 1 / 6 p (1 / 6)(5 / 6) / 8000 =- 2 . 05 <- 1 . 645, reject H . (c) [0 , b p + 1 . 645 p b p (1- b p ) / 8000] = [0 , . 1648], 1 / 6 = 0 . 1667 is not in this interval. This is consistent with the conclusion to reject H . 8.1–8 The value of the test statistic is z = . 70- . 75 p (0 . 75)(0 . 25) / 390 =- 2 . 280 . (a) Since z =- 2 . 280 <- 1 . 645, reject H . (b) Since z =- 2 . 280 >- 2 . 326, do not reject H . (c) p-value ≈ P ( Z ≤ - 2 . 280) = 0 . 0113. Note that 0 . 01 < p-value < . 05. 109 110 Chapter 8 8.1–10 (a) H : p = 0 . 14; H 1 : p > . 14; (b) C = { z : z ≥ 2 . 326 } where z = y/n- . 14 p (0 . 14)(0 . 86) /n ; (c) z = 104 / 590- . 14 p (0 . 14)(0 . 86) / 590 = 2 . 539 > 2 . 326 so H is rejected and conclude that the campaign was successful. 8.1–12 (a) z = y/n- . 65 p (0 . 65)(0 . 35) /n ≥ 1 . 96; (b) z = 414 / 600- . 65 p (0 . 65)(0 . 35) / 600 = 2 . 054 > 1 . 96, reject H at α = 0 . 025. (c) Since the p-value ≈ P ( Z ≥ 2 . 054) = 0 . 0200 < . 0250, reject H at an α = 0 . 025 significance level; (d) A 95% one-sided confidence interval for p is [0 . 69- 1 . 645 p (0 . 69)(0 . 31) / 600 , 1] = [0 . 659 , 1] . 8.1–14 We shall test H : p = 0 . 20 against H 1 : p < . 20. With a sample size of 15, if the critical region is C = { x : x ≤ 1 } , the significance level is α = 0 . 1671. Because x = 2, Dr. X has not demonstrated significant improvement with these few data. 8.1–16 (a) | z | = | b p- . 20 | p (0 . 20)(0 . 80) /n ≥ 1 . 96; (b) Only 5/54 for which z =- 1 . 973 leads to rejection of H , so 5% reject H . (c) 5%. (d) 95%. (e) z = 219 / 1124- . 20 p (0 . 20)(0 . 80) / 1124 =- . 43, so fail to reject H . 8.1–18 (a) Under H , b p = (351 + 41) / 800 = 0 . 49; | z | = | 351 / 605- 41 / 195 | s (0 . 49)(0 . 51) 1 605 + 1 195 = | . 580- . 210 | . 0412 = 8 . 99 . Since 8 . 99 > 1 . 96, reject H . (b) . 58- . 21 ± 1 . 96 r (0 . 58)(0 . 42) 605 + (0 . 21)(0 . 79) 195 . 37 ± 1 . 96 √ . 000403 + 0 . 000851 . 37 ± . 07 or [0 . 30 , . 44] ....
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ch8 - Chapter 8 Tests of Statistical Hypotheses 8.1 Tests...

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