EXAMPLE 9.15 The Sign Test as a Goodness-of-Fit Test

In Example 7.20 (page 407), we used a sign test to examine the effect of altering a software parameter on the measurement of complex machine parts. The study measured 76 machine parts, each with and without an option available in the software algorithm. The measurement was larger with the option on for 43 of the parts, and it was larger with the option off for the other 33 parts.

The sign test examines the null hypothesis that parts are equally likely to have larger measurements with the option on or off. Because , the sample proportion is and the null hypothesis is .

To look at these data from the viewpoint of goodness of fit, we think of the data as two counts: parts with larger measurements with the option on and parts with larger measurements with the option off.

If the two outcomes are equally likely, the expected counts are both . The expected counts are both greater than 5, so we can proceed with the significance test.

The test statistic is

We have , so the degrees of freedom are 1. From Table F we conclude that . The effect the option being on or off is not statistically significant.