Chapter 7: Parameter Estimation 39. The amount of beryllium in a substance is often determined by the use of a photometric filtration method. If the weight of the beryllium is u, then the value given by the photometric filtration method is normally distributed with mean u and standard deviation o. A total of eight independent measurements of 292 3.180 mg of beryllium gave the following results. 3.166, 3.192, 3.175, 3.180, 3.182, 3.171, 3.184, 3.177 Use the preceding data to (b) find a 90 percent confidence interval estimate of o. 40. If X1,...,X, is a sample from a normal population, explain how to obtain a 100(1-a) percent confidence interval for the population variance o when the population mean u is known. Explain in what sense knowledge of u improves (a) estimate o; the interval estimator compared with when it is unknown. Repeat Problem 38 if it is known that the mean burning time is 53.6 seconds. 41. A civil engineer wishes to measure the compressive strength of two differen s of concrete. A random sample of 10 specimens of the first type yielded the types following data (in psi) Type 1: 3,250, 3,268, 4,302, 3,184, 3,266 3,297, 3,332, 3,502, 3,064, 3,116 whereas a sample of 10 specimens of the second yielded the data Type 2: 3,094, 3,106, 3,004, 3,066, 2,984, 3,124, 3,316, 3,212, 3,380, 3,018 If we assume that the samples are normal with a common variance, determine (a) a 95 percent two-sided confidence interval for ui - u2, the difference in means; (b) a 95 percent one-sided upper confidence interval for u1 - u2; (c) a 95 percent one-sided lower confidence interval for u1 - 2. 42. Independent random samples are taken from the output of two machines on a production line. The weight of each item is of interest. From the first machine, a sample of size 36 is taken, with sample mean weight of 120 grams variance of 4. From the second machine, a sample of size 64 is taken, with a sample mean weight of 130 grams and a sample variance of 5. It is assumed that the weights of items from the first machine are normally distributed with mean lj and variance o and that the weights of items from the second machine are normally distributed with mean u2 and variance o2 (that is, the variances are assumed to be equal). Find a 99 percent confidence interval for u1 - H2, the and a sample g2 difference in population means.