A manufacturing company produces bearings. One line of bearings is specified to be 1.64 centimeters (cm) in diameter. A major customer requires that the variance of the bearings be no more than [tex]$0.001 \, \text{cm}^2$[/tex]. The producer is required to test the bearings before they are shipped. The diameters of 16 bearings are measured with a precise instrument, resulting in the following values:

1.69, 1.62, 1.63, 1.70, 1.66, 1.63, 1.65, 1.71, 1.64, 1.69, 1.57, 1.64, 1.59, 1.66, 1.63, 1.65

Assume bearing diameters are normally distributed. Use the data and [tex]$\alpha = 0.025$[/tex] to test whether the population of these bearings should be rejected because of too high variance.

To test for a high variance in the population of bearings, a chi-square test is used. The sample variance is compared to the critical value to determine whether to reject the null hypothesis. In this case, there is insufficient evidence to conclude that the population of bearings should be rejected due to a high variance.

Explanation:

Testing for Variance

To test whether the population of these bearings should be rejected due to a high variance, we can use a chi-square test. First, we need to calculate the sample variance and the critical value. The sample variance is calculated by summing the squared differences between each measurement and the mean, and then dividing by the sample size minus one. The critical value is obtained from the chi-square distribution table, using the significance level and the degrees of freedom.

Calculations

The sample variance is 0.002399 and the critical value is 26.296. Next, we compare the sample variance to the critical value. If the sample variance is greater than the critical value, we reject the null hypothesis that the population variance is less than or equal to the specified limit. In this case, the sample variance is less than the critical value, so we do not reject the null hypothesis. Therefore, there is insufficient evidence to conclude that the population of these bearings should be rejected because of a too high variance.

Keywords

variance, chi-square test, sample variance, critical value, null hypothesis

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