The continuity correction. One reason why the Normal approximation may fail to give accurate estimates of binomial probabilities is that the binomial distributions are discrete 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 approximation by treating each whole number count as if it occupied the interval from 0.5 below the number to 0.5 above the number. For example, approximate a binomial probability 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 13.30 that TigerWoods hits the fairway in 60% of his drives.

We will assume that his drives are independent 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.