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Unformatted text preview: Introduction to hypothesis tests – Examples (with solutions) 1. A grocery store had an average checkout time of 3 minutes. The management wished to improve this and installed a new checkout system. To test the new performance, they measured the checkout times x 1 ,...,x 942 of a sample of 942 customers who checked out at various times over the course of a week. The observed sample mean checkout time was ¯ x = 2 . 9 minutes and the observed sample standard deviation was s = 1 . 2 minutes. The management would like to know if these data provide statistical evidence at the α = 0 . 05 significance level that μ , the mean of the distribution of checkout time, is less than 3 minutes. (a) State the null and alternative hypotheses. Solution: H : μ = 3 H a : μ < 3 (b) What assumptions must be made about the observations x 1 ,...,x 942 to conduct the hypothesis test. Solution: We must assume that the measurements x 1 ,...,x 942 are a realization of a random sample from N ( μ,σ ); however, since the sample size n is very large, the assumption that this distribution is Normal could be relaxed, which would make this test approximate. (c) What is the rejection region for this test. Solution: This is a left-sided test with α = 0 . 05; we reject H if the test statistic realization t <- t 1- α,n- 1 =- t . 95 , 941 =- qt (0 . 95 , 941) =- 1 . 646475. (i.e. reject H if t <- 1 . 646475). (d) Compute the test statistic realization and p-value. t = ¯ x- μ s/ √ n = 2 . 9- 3 . 1 . 2 / √ 942 =- 2 . 5577 p- value = P ( T ≤ t ) = P ( T ≤ - 2 . 5577) = pt (- 2 . 5577 , 941) = 0 . 00534621 1 (e) State the conclusion to this hypothesis test. What type of error could we be(e) State the conclusion to this hypothesis test....
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- Spring '08