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Tut sol week9

# Tut sol week9 - BES Tutorial Sample Solutions S1/10 WEEK 9...

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1 BES Tutorial Sample Solutions, S1/10 WEEK 9 TUTORIAL EXERCISES (To be discussed in the week starting May 3) 1. Perform the following hypothesis tests of the population mean. In each case, illustrate the rejection regions on both the Z and X distributions, and calculate the p-value (prob-value) of the test. (a) H 0 : μ = 50, H 1 : μ > 50, n = 100, ݔ ҧ = 55, σ = 10, α = 0.05 Rejection region: ݖ ൌ ݔ ҧ െ 50 10 √100 ൐ ݖ ଴.଴ହ ൌ 1.645 Alternatively ݔ ҧ ൐ ݔ ҧ ൌ ߤ ൅ ݖ ଴.଴ହ ߪ ݊ ൌ 50 ൅ 1.645 ൬ 10 √100 ൰ ൌ 51.645 Since ݖ ൌ 55 െ 50 10 √100 ൌ 5 ൐ ݖ ଴.଴ହ ൌ 1.645 Can reject 0 H and conclude that the population mean is greater than 50 . 0.05 X 50 51.645 reject

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2 0.05 0 1.645 Z reject ݌ െ ݒ݈ܽݑ݁ ൌ ܲሺܼ ൐ 5ሻ ൎ 0.0000 (b) H 0 : μ = 25, H 1 : μ < 25, n = 100, ݔ ҧ = 24, σ = 5, α = 0.1 Rejection region: ݖ ൌ ݔ ҧ െ 25 5 √100 ൏ ݖ ଴.ଵ ൎ െ1.28 Alternatively ݔ ҧ ൏ ݔ ҧ ൌ ߤ െ ݖ ଴.ଵ ߪ ݊ ൌ 25 െ 1.28 ൬ 5 √100 ൰ ൌ 24.36 Since ݖ ൌ 24 െ 25 5 √100 ൌ െ2 ൏ െݖ ଴.ଵ ൌ െ1.28 Can reject 0 H and conclude that the population mean is less than 25 . 0.1 X 24.36 25 reject
3 0.1 ‐1.28 0 Z reject ݌ െ ݒ݈ܽݑ݁ ൌ ܲሺܼ ൏ െ2ሻ ൌ 0.0228 (c) H 0 : μ = 80, H 1 : μ 80, n = 100, ݔ ҧ = 80.5, σ = 4, α = 0.05 Rejection region: ݖ ൌ ݔ ҧ െ 80 4 √100 ൏ െݖ ଴.଴ଶହ ൎ 1.96 ݋ݎ ൐ 1.96 Alternatively: ݔ ҧ ൏ ݔ ҧ ൌ ߤ െ ݖ ଴.଴ଶହ ߪ ݊ ൌ 80 െ 1.96 ൬ 4 √100 ൰ ൌ 79.216

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