Question

Let \(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample of size 10 from a normal distribution \(N\left(0, \sigma^{2}\right) .\) Find a best critical region of size \(\alpha=0.05\) for testing \(H_{0}: \sigma^{2}=1\) against \(H_{1}: \sigma^{2}=2 .\) Is this a best critical region of size \(0.05\) for testing \(H_{0}: \sigma^{2}=1\) against \(H_{1}: \sigma^{2}=4 ?\) Against \(H_{1}: \sigma^{2}=\sigma_{1}^{2}>1 ?\)

Short answer

The optimal critical region for testing \(H_{0}: \sigma^{2}=1\) against the given alternatives at a level \(\alpha=0.05\) is when \(T > 16.92\). Whether this is the 'best' critical region for each alternative depends on the specific value of the alternative variance. For larger \(\sigma_{1}^{2}\), the critical region should be larger.

Step-by-step solution

Derive the test statistic

The variance of a normal distribution follows a chi-square distribution when multiplied by \(n-1\) (sample size minus 1). So, we first take the sample variance \(S^{2}\), which is calculated from the given samples \(X_{1}, X_{2}, \ldots, X_{10}\). Multiply \(S^{2}\) by \(n-1\), in this case 10-1=9, to form the test statistic, \(T=9S^{2} / \sigma_{0}^{2}\), where \(\sigma_{0}^{2}\) is the variance under the null hypothesis, \(H_{0}\). This is chi-square distributed with 9 degrees of freedom.

Determine the critical region

We are asked to find the critical region of size \(\alpha=0.05\). For a chi-square distribution with 9 degrees of freedom, we look up the value of \(k\), such that the probability of the chi-square statistic being more than \(k\) equals \(\alpha\). So, we want \(P(T > k)=0.05\). Using standard statistical tables, or a statistical software's chi-square calculator, we find that \(k \approx 16.92\). Our critical region is then \(T > 16.92\).

Test against different alternative hypotheses

If we want to test against \(\sigma^{2}=2\) (\(H_{1}\)), we must remember that a higher value for the alternative variance means that more extreme values are expected, making it harder to reject the null hypothesis. By contrast, an alternative variance of \(\sigma^{2}=4\) (\(H_{2}\)) means that extreme values are more likely, so the null hypothesis is easier to reject. Finally, if we test against \(\sigma^{2}=\sigma_{1}^{2}>1\) (\(H_{3}\)), the critical region would depend on the specific value of \(\sigma_{1}^{2}\), but would generally become larger for larger values of \(\sigma_{1}^{2}\).