Question: The data points in a sample of size 9 are 34, 51, 40, 38, 47, 50, 52, 44, 37.

a. If you can take the deviations of these data from any number you select, and you want to minimize the sum of the squared deviations (SSD), what number would you select? What is the minimized SSD? How many degrees of freedom are associated with this SSD? Calculate the mean squared deviation (MSD) by dividing the SSD by its degrees of freedom. (This MSD is an unbiased estimate of population variance.)

b. If you can take the deviations from three different numbers you select, and the first number is to be used with the first four data points to get the deviations, the second with the next three data points, and the third with the last two data points, what three numbers would you select? What is the minimized SSD? How many degrees of freedom are associated with this SSD? Calculate MSD.

c. If you can select nine different numbers to be used with each of the nine data points, what numbers would you select? What is the minimized SSD? How many degrees of freedom are associated with this SSD? Does MSD make sense in this case?

d. If you are told that the deviations are to be taken with respect to 50, what is the SSD? How many degrees of freedom are associated with this SSD? Calculate MSD.