Question

Specifications for mass-produced bearings of a certain type require among other things, that the standard deviation of their outside diameters should not exceed 0.0050cm. Use the level of significance 0.01 to test the null hypothesis σ=0.0050 against the alternative hypothesis σ>0.0050 on the basis of a random sample of n=12 for which s=0.0077cm. What would be the decision for the test of hypothesis for this problem?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\sigma^2=0.000025$

$H_1:\sigma^2>0.000025$

This corresponds to a right-tailed test test, for which a Chi-Square test for one population variance will be used.

Based on the information provided, the significance level is $\alpha = 0.01,$ $df=n-1=12-1=11$ degrees of freedom, and the the rejection region for this right-tailed test is$R = \{\chi^2: \chi^2 > 24.725\}.$

The Chi-Squared statistic is computed as follows:

$$\chi^2=\dfrac{(n-1)s^2}{\sigma^2}=\dfrac{(12-1)(0.0077)^2}{(0.005)^2}=26.0876$$

Since it is observed that $\chi^2=26.0876>24.725=\chi_c^2,$  it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population standard deviation $\sigma$ is greater than $0.005$, at the $0.01$ significance level.