Statistics 101-106, Fall 1998

Pollard homework sheet 5

Due: Thursday 15 October

Your task for this week is to draw a histogram that shows, with "reasonable accuracy", the distribution of the median for samples of size 11 taken from the standard normal distribution. The histogram will be based on n samples of size 11, for a value of n that you will determine.

I want the bins for the histogram to be of width 1/4, and I want the area under the histogram to equal 1. I would like to be reasonably sure (say, with probability 0.95) that the height of the central bar (the one for the bin from 0 to 1/4) lies within 5% of its true, theoretical value.

  1. Experiment with various values of n to get a rough idea of the probability, p, that the median of a sample of size 11 lies between 0 and 1/4.
  2. What should the true height, h, of the central bar be, roughly? (Use your rough estimate of p from part i.)
  3. What is the approximate distribution of the height, H, of the central bar that you will get from n samples of size 11?
  4. Roughly, how large must n be in order to ensure that
    2 standard deviations of H < h/20
  5. Draw the histogram, using an appropriate n.
  6. What is your guess for the true value h?

    You might find it convenient to carry out your experiments by means of a macro. If so, you should use a text editor to create a file called myhist.mac in the T: directory (or in some other directory where you can save things). The file should contain lines:

    [put your commands for drawing the histogram in here]
    To get Minitab to run your commands you would type
    at the Minitab prompt (in the session window--don't forget to enable commands first).

    If you want to experiment using the menus, look in the history window for the corresponding commands. You can copy and paste these commands into your macro.

    Each Section leader will assign extra problems.