. Consider drawing a random sample of size n = 100 of a random variable X that has a Bernoulli distribution with probability of success p = .75.

(a) Let p be the sample proportion, that is the sample statistic which reports the proportion of successes out of 100 trials. Note that p = X n , with X counting the number of successes in the sample. What is the expected value of the sample proportion, p? As a hint, determine the distribution of the random variable X. Then use properties of this distribution as well as the rule for expected value of a constant 1 n times a random variable (X).

(b) What is the variance of p? Use the same process -determine the distribution of the random variable X. Then use properties of this distribution as well as the rule for variance of a constant 1 n times a random variable (X).

(c) Use the Central Limit Theorem to calculate the following probability: P(p .98)?

(d) Use the Central Limit Theorem to calculate the following probability: P(p .02)?

(e) Why is it valid to use the Central Limit Theorem?