22.28 Finding a P-value by simulation. A classic experiment to detect extra-sensory perception (ESP) uses a shuffled deck of cards containing five suits (waves, stars, circles, squares, and crosses). As the experimenter turns over each card and concentrates on it, the subject guesses the suit of the card. A subject who lacks ESP has probability 1-in-5 of being right by luck on each guess. A subject who has ESP will be right more often. Julie is right in five of 10 tries. (Actual experiments use much longer series of guesses so that weak ESP can be spotted. No one has ever been right half the time in a long experiment!)

  1. (a) Give H0 and Ha for a test to see if this result is significant evidence that Julie has ESP.

  2. (b) Explain how to simulate the experiment if we assume for the sake of argument that H0 is true.

  3. (c) Simulate 20 repetitions of the experiment; begin at line 121 of Table A.

  4. (d) The actual experimental result was five correct in 10 tries. What is the event whose probability is the P-value for this experimental result? Give an estimate of the P-value based on your simulation. How convincing was Julie’s performance?

The following exercises concern the optional section on calculating P-values. To carry out a test, complete the steps (hypotheses, sampling distribution, data, P-value, and conclusion) illustrated in Example 3.