A random sample of n= 15 heat pumps of a certain type yielded the following observations on lifetime (in years):2.0 1.1 6.0 1.915.6 0.9 4.8 0.9 12.1 5.3 0.65.4 0.4 1.0 5.3(a) Assume that the lifetime distribution is exponential and use an argument parallel to that of this example to obtain a95% CI for expected (true average) lifetime. (Round your answers to two decimal places.)years(b) How should the interval of part (a) be altered to achieve a confidence level of 99%?O A 99% confidence level requires using a new value of n to capture an area of 0.1 in each tail of the chi-squareddistribution.O A 99% confidence level requires using critical values that capture an area of 0.1 in each tail of the chi-squareddistribution.O A 99% confidence level requires using critical values that capture an area of 0.005 in each tail of the chi-squareddistribution.O A 99% confidence level requires using a new value of n to capture an area of 0.005 in each tail of the chi-squareddistribution.(c) What is a 95% CI for the standard deviation of the lifetime distribution? [Hint: What is the standard deviation of anexponential random variable?] (Round your answers to two decimal places.)years

Sample size Sample data21.161.95.40.415.315.60.94.80.912.15.30.6A confidence interval includes a range of values within which the population parameter is expected to lie.Confidence level is 95% i.e., 0.95.Significance level This implies: and The distribution of lifetime of heat pumps of a certain type follow exponential distribution.Let X denote the lifetime distribution. Then, X follows Exponential distribution with mean .This implies:Here, is the true average lifetime.Now let follows gamma distribution with scale 1 and shape 15. Then, confidence interval for is expressed as: Use Excel function "=GAMMA.INV(0.025,15,1)" to obtain the value of . The result is shown below:Use Excel function "=GAMMA.INV(0.975,15,1)" to obtain the value of . The result is shown below:Calculate as follows:Thus, we have:Therefore, the 95% CI for expected true average lifetime is years.(b) For 99% confidence level, the significance level is .This implies: A 99% confidence interval for expected true average lifetime is obtained by using the critical value beyond which the area is 0.005 in each tail of the Chi-square distribution, where Chi-square distribution is a special case of gamma distribution when and .Therefore, the correct statement is:A 99% confidence level requires using critical values that capture an area of 0.005 in each tail of the chi-squared distribution.(c) We know that, the mean and standard deviation for exponential distribution are the same.Thus, the 95% CI for standard deviation of true average lifetime is years.(a) The 95% CI for expected true average lifetime is years.(b) A 99% confidence level requires using critical values that capture an area of 0.005 in each tail of the chi-squared distribution.(c) The 95% CI for standard deviation of true average lifetime is years.