Chapter 6: Q77E (page 364)

USGA golf ball tests. The United States Golf Association (USGA) tests all new brands of golf balls to ensure that they meet USGA specifications. One test conducted is intended to measure the average distance traveled when the ball is hit by a machine called "Iron Byron," a name inspired by the swing of the famous golfer Byron Nelson. Suppose the USGA wishes to estimate the mean distance for a new brand to within 1 yard with 90% confidence. Assume that
past tests have indicated that the standard deviation of the distances Iron Byron hits golf balls is approximately 10 yards. How many golf balls should be hit by Iron Byron to achieve the desired accuracy in estimating the mean?

Short Answer

Expert verified

Iron Byron needs hit 271 golf balls to reach the desired reliability in determining the mean.

Step by step solution

Given information

The USGA wants to estimate the average distance for a new brand to within one yard with 90 percent certainty. Suppose that previous tests have revealed that the normal variation of Iron Byron's golf ball distances is roughly 10 yards

Finding the sample size

The standard error is 1 in this case.

The standard deviation value is 10.

A 90% confidence interval's critical value is $z_{α / 2} = z_{0.10 / 2} = z_{0.05} = 1.645$

So, the standard error is,

$\begin{array}{rcl} S E & = & z_{α / 2} \frac{σ}{\sqrt{n}} \\ n & = & {z^{2}}_{α / 2} \frac{σ^{2}}{S E^{2}} \\ n & = & 1 . 645^{2} \times \frac{10^{2}}{1^{2}} \\ n & = & 270.6025 \\ n & ≃ & 271 \end{array}$