Problem 1E The true height of a mountain peak is unknown. Repeated measurements of the height are made using a laser altimeter. The measurements X1, X2,. . . ,Xn satisfy the following conditions: for 1 S i S n, X=pb+6 where a is the unknown constant height of the mountain peak, and 6,; is a random error. The errors are i.i.d. with mean 0 and unknown SD 0. The altimeter is used to make 100 measurements. The measurements have an average of 75, 348 inches and an SD of 25 inches. a) Construct an approximate 95% condence interval for ,u. b) The interval in part a) is constructed (i) to estimate the average of the 100 measurements and give ourselves some room for error in the estimate. (ii) to estimate the height of the mountain and give ourselves some room for error in the estimate. (iii) to provide a range in which 95 of the 100 measruments are likely to have fallen. (iv) to provide a range in which 95% of all possible measurements are likely to fall. Which of (1') (in) are false? Explain why they are false. c) If possible, sketch the histogram of the 100 measurements, justify your choice of shape, and indicate the mean and the SD of the hitogram as clearly as you can. If it is not possible to sketch the histogram with the information given, explain why not. d) A new set of measurements will be made using this altimeter. Approximately how large should the set be to have about a 95% chane that its average is within 1 inch of u