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Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 09.07.12 /EP All rights reserved. Legal action will be taken against infr ingement. Reproduction is prohibited without prior consent. Name: Student Id. Number: Properly justify all your answers. Explicitly define all the random variables you are going to use in the solution which have not already been introduced in the text. Exercise 1 There is medical interest in investigating if sport activit y leads to a decrease in the resting heart rate (HRrest). In order to check such a claim, 8 volunteers who never did exercise before started a one-month jogging program. Data on the volu nteers’ resting heart rate before and after the month of exercise are as follows: X HRrest before 74 86 98 102 78 84 79 70 Y HRrest after 69 84 89 109 70 79 68 73 Assume that the data are Gaussian. 1. Based on the available data, construct a bilateral 99% con fidence interval for the difference between the mean heart rate after and before the month of jogg ing. 2. Can we conclude, at the 10% significance level, that sport a ctivity reduces the mean resting heart rate? In addition, find the p-value of this test. 3. At the same significance level as at point 2., is there evide nce to conclude that the mean resting heart rate has changed? Solution Data in the table are observations from a paired bivariate Gaussian sample (X1, Y1),. . ., (X8, Y8). Let W be the r.v. that represents the difference between the heart rate after and before the sport activity, W = Y − X. A sample of size 8 for W is obtained: ( − 5, − 2, − 9, 7, − 8, − 5, − 11, 3), and ¯w8 = − 3.75, s2 W = 37 .92857. Denote by µW = µY − µX the mean difference between the heart rates. 1. Since tα/2,n−1 = t0.005,7 = 3.499, the 99% confidence interval for µW is ( ¯wn −…