Question

Suppose that the thickness of a part used in a semiconductor is its critical dimension and that measurements of the thickness of a random sample of 18 such parts have the variance s2 = 0.68,
where the measurements are in thousandths of an inch. The process is considered to be under control if the variation of the thickness is given by a variance not greater than 0.36. Assuming
that the measurements constitute a random sample from a normal population, test the claim at the
α =0.05 significance level.

Answer 1

The null hypothesis: H₀: $\sigma^2 <= 0.36$

The alternative hypothesis: Ha: $\sigma^2 > 0.36$

For 0.05 significance level and (n-1 = 17) degrees of freedom; critical value of $\chi^2 = 27.587$

Decision rule: reject Ho if the test statistic $\chi^2 > 27.587$

The test statistic

$\chi^2 =\frac{(n-1) \times s^2}{\sigma^2}$

$\chi^2 = \frac{17 \times 0.68}{0.36} = 32.111$

As test statistic is higher than the critical value we reject the null hypothesis.

So, we have sufficient evidence to conclude at 0.05 significance level that the population variance is greater than 0.36 level and process is out of control.