5.79 Is the ESP result better than guessing? When the ESP study of Exercise 5.77 discovers a subject whose performance appears to be better than guessing, the study continues at greater length. The experimenter looks at many cards bearing one of five shapes (star, square, circle, wave, and cross) in an order determined by random numbers. The subject cannot see the experimenter as the experimenter looks at each card in turn, in order to avoid any possible nonverbal clues. The answers of a subject who does not have ESP should be independent observations, each with probability 1/5 of success. We record 900 attempts.

1. (a) What are the mean and the standard deviation of the count of successes?

2. (b) What are the mean and the standard deviation of the proportion of successes among the 900 attempts?

3. (c) What is the probability that a subject without ESP will be successful in at least 24% of 900 attempts?

4. (d) The researcher considers evidence of ESP to be a proportion of successes so large that there is only probability 0.01 that a subject could do this well or better by guessing. What proportion of successes must a subject have to meet this standard? (Example 1.45, on pages 65–66, shows how to do an inverse calculation for the Normal distribution that is similar to the type required here.)