Twenty motors were put on test under a high-temperature setting. The lifetimes in hours of the motors under these conditions are given below. Suppose we assume that the lifetime of a randomly selected motor under these conditions, X, has a Gamma (1, theta) distribution.
(a) Derive the maximum likelihood estimate of 9 and evaluate it numerically.
(b) Obtain the sample median of the data, which is an estimate of the median lifetime of a motor. What parameter is it estimating (i. e., determine the median of X)?
(c) Based on the mle, what is another estimate of the median of X?
(d) Obtain a large sample approximate 95% confidence interval for the mean lifetime of a motor.