Suppose that in the general population of men between 65 and 70 years old, sixty percent have a resting systolic blood pressure over 120 mm Hg, considered a high blood pressure. Consider a sample of twenty men randomly sampled from this population. For each one, it is determined whether or not they have high blood pressure.
a. What is the probability distribution of the number of men in the sample who have high blood pressure? Give the name of the distribution and the numerical values of the parameters it depends on (e.g. the parameters the Normal distribution depends on are the mean and the standard deviation). Explain the conditions that must be satisfied to justify using this distribution, and comment on whether or not they are satisfied.
b. Give the mean and standard deviation of the distribution in part a, and show how they are calculated.
c. Calculate the probability that from 10 to 12 of the men (i.e. 10, 11, or 12 men) have high blood pressure. Explain how the calculation is done, show the formulas used for the calculation, and give the numerical result.
d. Use a Normal approximation to the probability distribution in part a., to obtain an approximate answer to part c. Specify the best interval to use for the probability of the Normal random variable. In other words, what is the interval of the best area under the Normal bell curve to use to get the answer? Hint: Where would the interval for the probability of each number (10, 11, or 12) be centered, and how wide would it be? Compare your result to the exact answer obtained in part c.