Question

(14.43) 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 probabilityP(X⩾10)P(X⩾10)by finding the Normal probabilityP(X⩾9.5)P(X⩾9.5). Be careful: binomialP(X>10)P(X>10)is approximated by NormalP(X⩾10.5)P(X⩾10.5). Adding 0.5 to the length of the interval is called continuity correction. One statistic used to assess professional golfers is driving accuracy, the percent of drives that land in the fairway. In 2013, driving accuracy for PGA Tour professionals ranged from about 45% to about 75%. Tiger Woods, the highest money winner on the PGA tour in 2013, only hits the fairway about 57 % of the time. We will assume that his drives are independent and that each has probability 0.57 of hitting the fairway. Suppose Woods drives 24 times. (a)Does this setting satisfies the rule of thumb for use of the Normal approximation (just barely) No Yes (b) The exact binomial probability that he hits 17 or more fairways is (±±0.001) (c) What is the Normal approximation (±±0.0001 (use software)) to P(X⩾17)P(X⩾17) without using continuity correction? (d) What is the Normal approximation (±±0.0001 (use software)) to P(X⩾17)P(X⩾17) using the continuity correction?

Answer 1