A manufacturing process drills holes in sheet metal that are...

A manufacturing process drills holes in sheet metal that are supposed to be $.5000 \mathrm{~cm}$ in diameter. Before and after a new drill press is installed, the hole diameter is carefully measured (in cm) for 12 randomly chosen parts. At $\alpha=.05$, do these independent random samples prove that the new process has smaller variance? Show the hypotheses, decision rule, and test statistic. $$ \begin{array}{lllllll} \hline \text { New drill: } & .5005 & .5010 & .5024 & .4988 & .4997 & .4995 \\ & .4976 & .5042 & .5014 & .4995 & .4988 & .4992 \\ \text { Old drill: } & .5052 & .5053 & .4947 & .4907 & .5031 & .4923 \\ & .5040 & .5035 & .5061 & .4956 & .5035 & .4962 \\ \hline \end{array} $$

A manufacturing process drills holes in sheet metal that are supposed to be $.5000 \mathrm{~cm}$ in diameter. Before and after a new drill press is installed, the hole diameter is carefully measured (in cm) for 12 randomly chosen parts. At $\alpha=.05$, do these independent random samples prove that the new process has smaller variance? Show the hypotheses, decision rule, and test statistic.
$$
\begin{array}{lllllll}
\hline \text { New drill: } & .5005 & .5010 & .5024 & .4988 & .4997 & .4995 \\
& .4976 & .5042 & .5014 & .4995 & .4988 & .4992 \\
\text { Old drill: } & .5052 & .5053 & .4947 & .4907 & .5031 & .4923 \\
& .5040 & .5035 & .5061 & .4956 & .5035 & .4962 \\
\hline
\end{array}
$$

Key Concepts

Decision Rule

The decision rule in this testing context involves comparing the calculated F test statistic with the critical value from the F-distribution at a specified significance level (?). If the test statistic falls in the rejection region—typically, if it is less than the critical value for a one-tailed test—the null hypothesis is rejected in favor of the alternative hypothesis, indicating that the new process has a significantly smaller variance. The significance level (?) defines the probability of rejecting the null hypothesis when it is actually true.

Test Statistic

The test statistic in an F-test is computed as the ratio of the two sample variances. When testing whether one variance is smaller than the other, the variance from the process claimed to be more consistent (smaller variance) is placed in the numerator and the variance from the process with presumed higher variability is in the denominator. This ratio is then compared to a critical value derived from the F-distribution.

Null and Alternative Hypotheses

In hypothesis testing for variance, the null hypothesis (H0) typically states that there is no difference in the variability of the two processes, meaning the variance of the new process is equal to the variance of the old process. The alternative hypothesis (Ha) in this case is one?sided; it asserts that the variance of the new drill process is smaller than that of the old process, reflecting the manufacturer's improvement claim.

F-Test for Comparing Variances

The F-test is a statistical method used to compare the variances of two independent samples. By calculating the ratio of the sample variances, this test determines whether the observed difference between the variances is statistically significant. The test is based on the F-distribution which is appropriate when the samples come from normally distributed populations.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis. In the context of the F-test for variances, each sample contributes its own degrees of freedom (n - 1) where n is the sample size. These degrees of freedom are used to determine the appropriate critical values from the F-distribution for the given test.

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