c. Determine the critical values for a two-tailed test of a population standard deviation for a sample of size n = 30 at the α = 0.05 level of significance.

Verified step by step guidance

1. Identify the type of test and the distribution to use: Since we are testing the population standard deviation and the sample size is 30, we use the Chi-square distribution for the test statistic.
2. Determine the degrees of freedom (df) for the Chi-square distribution, which is given by \(df = n - 1\). For \(n = 30\), calculate \(df = 30 - 1 = 29\).
3. Since this is a two-tailed test at the \(\alpha = 0.05\) significance level, split the significance level into two tails: \(\alpha/2 = 0.025\) for each tail.
4. Find the critical values from the Chi-square distribution table corresponding to \(df = 29\) at the lower tail probability of \$0.025\( and the upper tail probability of \)1 - 0.025 = 0.975$. These values are the critical values for the test.
5. Express the critical values as \(\chi^2_{\alpha/2, df}\) for the lower critical value and \(\chi^2_{1 - \alpha/2, df}\) for the upper critical value, which will be used to decide whether to reject the null hypothesis.

Key Concepts

Chi-Square Distribution

The chi-square distribution is used to test hypotheses about population variances or standard deviations. It is a right-skewed distribution that depends on degrees of freedom, which for variance tests is typically n - 1, where n is the sample size.

Two-Tailed Test

A two-tailed test evaluates whether a parameter is significantly different from a hypothesized value in either direction. The significance level α is split between the two tails of the distribution, so each tail has an area of α/2.

Critical Values

Critical values are the cutoff points on the test distribution that define the rejection regions for the null hypothesis. For a chi-square test on variance, these values are found using the chi-square distribution table at α/2 and 1 - α/2 with n - 1 degrees of freedom.