A machine produces ball bearings that are designed to have a diameter standard deviation of 0.04 mm, but an engineer suspects the variability has increased. A sample of 60 bearings shows a standard deviation of 0.052 mm. Perform a hypothesis test with $\alpha =0.01$ to test the claim. Should the line manager have the machine serviced?

• A. Since $P$-value $=7.59\times10^{-4}<0.01=\alpha$, we fail to reject $H_0$, not enough evidence to suggest $\sigma>0.04$.No need to service the machine.
• B. Since $P$-value $=0.0016<0.01=\alpha$, we fail to reject $H_0$, not enough evidence to suggest $\sigma>0.04$.No need to service the machine.
• C. Since $P$-value $=7.59\times10^{-4}<0.01=\alpha$, we reject $H_0$, enough evidence to suggest $\sigma>0.04$.Machine should be serviced.
• D. Since $P$-value $=0.0016<0.01=\alpha$, we reject $H_0$, enough evidence to suggest $\sigma>0.04$.Machine should be serviced.

Answer: (C)

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. Since the engineer suspects the variability has increased, set the null hypothesis as \(H_0: \sigma = 0.04\) mm (the standard deviation is as designed) and the alternative hypothesis as \(H_a: \sigma > 0.04\) mm (the standard deviation has increased).
2. Step 2: Identify the significance level \(\alpha = 0.01\) and the sample size \(n = 60\). Since the population variance is unknown and the sample size is large, use the chi-square test for variance.
3. Step 3: Calculate the test statistic using the formula for testing variance: \(\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}\) where \(s = 0.052\) mm is the sample standard deviation, \(\sigma_0 = 0.04\) mm is the hypothesized standard deviation, and \(n = 60\) is the sample size.
4. Step 4: Determine the critical value from the chi-square distribution with \(n-1 = 59\) degrees of freedom at the \(\alpha = 0.01\) significance level for a right-tailed test. This critical value will be the cutoff to decide whether to reject \(H_0\).
5. Step 5: Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis, indicating that the variability has increased and the machine should be serviced. Otherwise, do not reject \(H_0\).