Solution 3 - IE4230 Homework 3 1 In statistical hypothesis...

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IE4230 Homework 3 03/02/2015
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1. In statistical hypothesis testing, what are the type I and type II errors? What is the P -value? A type I error is the incorrect rejection of a true null hypothesis. A type II error is the failure to reject a false null hypothesis. The P -value is the probability that the test statistic will take on a value that is at least as extreme as the observed value of the statistic when the null hypothesis H 0 is true . Null hypothesis ( H 0 ) True False Reject null hypothesis Type I error Correct outcome Fail to reject null hypothesis Correct outcome Type II error
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2. The inside diameters of bearings used in an aircraft landing gear assembly are known to have a standard deviation of σ = 0.002 cm. A random sample of 15 bearings has an average inside diameter of 8.2535 cm. a) Test the hypothesis that the mean inside bearing diameter is 8.25 cm. Use a two-sided alternative and α = 0.05. The appropriate hypotheses are H 0 : μ = 8.25 H 1 : μ 8.25. Since the population standard deviation is known , the value of the test statistic is 𝑍 = 𝑥 15 − 𝜇 0 𝜎 𝑛 = 8.2535 − 8.25 0.002 15 ≅ 6.7777. Because Z > Z α /2 = 1.96, we reject H 0 : μ = 8.25 and conclude that the mean inside bearing diameter is not 8.25 cm.
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b) Find the P -value for this test. From Appendix Table II, we find the P -value for this test is 2 1 − Φ 6.7777 = 0. c) Construct a 95% two-sided confidence interval on mean bearing diameter. Because 𝑥 − 𝜇 𝜎 𝑛 ~𝑁 0,1 , the 100(1- α )% two-sided confidence interval can be computed by 𝑥 − 𝑍 𝛼 2 𝜎 𝑛 ≤ 𝜇 ≤ 𝑥 + 𝑍 𝛼 2 𝜎 𝑛 .
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