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mat2378-assignment4-with_sol

# mat2378-assignment4-with_sol - Assignment 4 Due date 16...

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Assignment 4 Due date: 16 November 2009 Total number of points: 27 Q 1. (6.39) In a natural population of mice near Ann Arbor, Michigan, the coats of some individuals are white-spotted on the belly. In a sample of 580 mice from the population, 28 individuals were found to have white-spotted nellies. Construct 95% confidence interval for the population proportion of this trait. Solution to Q1: ˆ p = 28 / 580 = 0 . 048, estimated standard error= 0 . 048 * (1 - 0 . 048) / 580 = 0 . 0088, so that the 95% confi- dence interval is 0 . 048 ± 1 . 96 * 0 . 0088 . Marking scheme for Q1: 2 points (1 point for the appropriate formula, 1 for the correct final answer). Q 2. Nine measurements of ozone concentration are obtained during a year: 3.5 5.1 6.6 6.0 4.2 4.4 5.3 5.6 4.4 Assuming the measurements come from a normal population, provide two 95% confidence intervals for the population mean μ : (a) one assuming that the population variance σ 2 = 1 . 21; (b) another relaxing this assumption . (c) For the situation in part (b), test H 0 : μ = 5 . 74 against H A : μ = 5 . 74 taking α = 0 . 05. (You have to use confidence intervals to answer this here). Solution to Q2: First, α = 1 - 0 . 95. Second, the sample mean is ¯ y = 5 . 01 and the sample standard deviation is s = 0 . 9765. (a) Assuming that the population variance is 1.21: 5 . 01 ± (1 . 96 × 1 . 21 / 9) 5 . 01 ± 0 . 72 or (4 . 29 , 5 . 73) . (b) If we relax the assumption, we have to compute ¯ y ± t α/ 2 ,n - 1 s n = 5 . 01 ± 2 . 306 0 . 9765 9 . Indeed, from tables we have that P ( t 8 > 2 . 306) = 0 . 025 P ( t 8 < - 2 . 306) = 0 . 025 Thus, t 0 . 025 , 8 = 2 . 306. The confidence interval is 5 . 01 ± 2 . 306 0 . 97 9 or (4 . 26 , 5 . 76) .

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mat2378-assignment4-with_sol - Assignment 4 Due date 16...

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