12.38 The continuity correction. One reason why the Nor mal approximation may fail to give accurate estimates of bi nomial probabilities is that the binomial distributions are dis crete and the Normal distributions are continuous. That is, counts take only whole number values but Normal variables can take any value. We can improve the Normal approxi mation by treating each whole number count as if it occu pied the interval from 0.5 below the number to 0.5 above the number. For example, approximate a binomial probabil ity P (X > 10) by finding the Normal probability P(X > 9.5).

Be careful: binomial P(X > 10) is approximated by Normal P(X> 10.5).

We saw in Exercise 12.27 that Tiger Woods hits the fairway in 60% of his drives. We will assume that his drives are inde pendent and that each has probability 0.6 of hitting the fairway.

Tiger drives 25 times. The exact binomial probability that he hits 15 or more fairways is 0.5858.

(a) Show that this setting satisfies the rule of thumb for use of the Normal approximation (just barely).

(b) What is the Normal approximation to P(X > 15)?

(c) What is the Normal approximation using the continuity correction? That's a lot closer to the true binomial probability.