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.

For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius and height 2r minus the volume of two cones, each with a radius and height of r. a cross section of the sphere is and a cross section of the cylinder minus the cones, taken parallel to the base of cylinder, is the volume of the cylinder with radius r and height 2r is and the volume of each cone with radius r and height r is 1/3 pie r^3. so the volume of the cylinder minus the two cones is therefore, the volume of the cylinder is 4/3pie r^3 by cavalieri's principle. (fill in options are: r/2- r- 2r- an annulus- a circle -1/3pier^3- 2/3pier^3- 4/3pier^3- 5/3pier^3- 2pier^3- 4pier^3)