Poblems 287 Tat: Make use of the formula for the moment generating function of a normal random variable. A The following are, in minutes, travel times to work over a sequence of 10 days. 42, 28, 53, 57, 67, 39, 35, 50, 44, 39 Assuming an underlying lognormal distribution, use the data to estimate the mean travel time. 8. An electric scale gives a reading equal to the true weight plus a random error that is normally distributed with mean 0 and standard deviation o = .1 mg. Suppose that the results of five successive weighings of the same object are as follows: 3.142, 3.163, 3.155, 3.150, 3.141. (a) Determine a 95 percent confidence interval estimate of the true weight. (b) Determine a 99 percent confidence interval estimate of the true weight. 9. The PCB concentration of a fish caught in Lake Michigan was measured by a technique that is known to result in an error of measurement that is normally distributed with a standard deviation of .08 ppm (parts per million). Suppose the results of 10 independent measurements of this fish are 11.2, 12.4, 10.8, 11.6, 12.5, 10.1, 11.0, 12.2, 12.4, 10.6 (a) Give a 95 percent confidence interval for the PCB level of this fish. (b) Give a 95 percent lower confidence interval. (c) Give a 95 percent upper confidence interval. 10. The standard deviation of test scores on a certain achievement test is 11.3. If a random sample of 81 students had a sample mean score of 74.6, find a 90 percent confidence interval estimate for the average score of all students. 11. Let X1,...,X, X+1 be a sample from a normal population having an unknown mean u and variance 1. Let Xn = E X;/n be the average of the first n of them. %3D %3D1 (a) What is the distribution of Xn+1- Xn? (b) If X = 4, give an interval that, with 90 percent confidence, will contain the value of X+1. 12. If X1,..., X, is a sample from a normal population whose mean u is unknown but whose variance o? is known, show that (-0, X+ Z0//n) is a 100(1- a) percent lower confidence interval for u. . A sample of 20 cigarettes is tested to determine nicotine content and the average value observed was 1.2 mg. Compute a 99 percent two-sided confidence inter- vai for the mean nicotine content of a cigarette if it is known that the standard deviation of a cigarette's nicotine content is o .2 mg. %3D