Can you solve these problems. Please skip as too long if you want to add bonus. A random sample of heat pumps of n = 15a certain type yielded the following observations on lifetime (in years): 2.0 1.3 6.0 1.9 5.1 4 1.0 5.3 15.7 .7 4.8 .9 12.2 5.3 .6 a. Assume that the lifetime distribution is exponential and use an argument parallel to that of Example 7.5 to obtain a 95% Cl for expected (true average) lifetime. b. How should the interval of part (a) be altered to achieve a confidence level of 99%? c. What is a 95% CI for the standard deviation of the lifetime distribution? [Hint. What is the standard deviation of an exponential random variable?] Determine the confidence level for each of the following large-sample one-sided confidence bounds: a. Upper bound: x + .84s/ Vn b. Lower bound: x - 2.05s/ Vn c. Upper bound: x + .67s/ Vn The article "Evaluating Tunnel Kiln Performance" (Amer. Ceramic Soc. Bull., Aug. 1997: 59-63) gave the following summary information for fracture strengths (MPa) of n = 169 ceramic bars fired in a particular kiln: x = 89.10. = 89.10 a. Calculate a (two-sided) confidence interval for true average fracture strength using a confidence level of 95%. Does it appear that true average fracture strength has been precisely estimated? b. Suppose the investigators had believed a prior that the population standard deviation was about 4 MPa. Based on this supposition, how large a sample would have been required to estimate u to within .5 MPa with 95% confidence