hwk5_sol - ISOM 111 L6-L7 Fall 2009 1 Homework 5 Solutions...

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Unformatted text preview: ISOM 111 L6-L7, Fall 2009 1 Homework 5 Solutions I . Problems 7.40 (p.281) of the textbook; and (c): Based on your 95% confidence interval found in Part (a), find the sample size needed so that the margin of error at 95% confidence will be no more than 2%. Solution : (a) A point estimate of the population proportion p is the sample proportion ˆ p = 410 / 1000 = 0 . 41. A 95% CI for p is [ˆ p ± z . 025 r ˆ p (1- ˆ p ) n = [0 . 41 ± 1 . 96 r . 41(1- . 41) 1000 ] = [0 . 3795 , . 4405] . (b) Based on the 95% CI, we are 95% confident that p is at least 0.3795 = 37.95%, hence a reasonable estimate of the minimum percentage of first-year defaults that are approved on the basis of falsified applications is 37.95%. (c) Again, based on the 95% CI, we are 95% confident that p is at most 0.4405, so the sample size needed so that the margin of error at 95% confidence will be no more than 2% is n = » . 4405(1- . 4405) · ‡ z . 025 . 02 · 2 … = 2367 . II . Problem 8.48 (p.318) of the textbook (let the significance level α be 0.01). Solution : (a) We’re testing a “not equal to” alternative when σ is known. The observed test statistic z = ¯ x- μ σ/ √ n = 3 . 3- 4 . 71 / √ 800 ≈ - 27 . 89 . So the p-value p = P ( | Z | ≥ | z | ) = P ( | Z | ≥ 27 . 89) < P ( | Z | ≥ 3 . 09) ≈ . 002 . Since the p-value is smaller than α = 0 . 01, we reject the null hypothesis at 1% significance level. And since the sample mean ¯ x = 3 . 3 < 4, we estimate that the true population mean is smaller than 4. (b) Similarly as in Part (a), the observed test statistic z = ¯ x- μ σ/ √ n = 4 . 3- 4 . 66 / √ 500 ≈ 10 . 16 , ISOM 111 L6-L7, Fall 2009 2 and the p-value p = P ( | Z | ≥ | z | ) = P ( | Z | ≥ 10 . 16) < P ( | Z | ≥ 3 . 09) ≈ . 002 . This is smaller than α = 0 . 01, so we reject the null hypothesis at 1% significance level. Finally, since the sample mean ¯ x = 4 . 3 > 4, we estimate that the true population mean is bigger than 4....
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This note was uploaded on 01/01/2010 for the course ISOM ISOM111 taught by Professor Anthonychan during the Spring '09 term at HKUST.

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hwk5_sol - ISOM 111 L6-L7 Fall 2009 1 Homework 5 Solutions...

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