b. Determine the critical value for a left-tailed test of a population standard deviation for a sample of size n = 23 at the α = 0.1 level of significance.

Verified step by step guidance

1. Identify the type of test and the distribution to use: Since this is a test about a population standard deviation and the sample size is small (n = 23), the test statistic follows a Chi-square distribution.
2. Determine the degrees of freedom (df) for the Chi-square distribution, which is calculated as \(df = n - 1\). For this problem, \(df = 23 - 1 = 22\).
3. Since it is a left-tailed test at the significance level \(\alpha = 0.1\), you need to find the critical value \(\chi^2_{\alpha, df}\) such that the area to the left of this value is 0.1.
4. Use a Chi-square distribution table or statistical software to find the critical value corresponding to \(df = 22\) and a cumulative probability of 0.1 (left tail).
5. This critical value will be the cutoff point for the test statistic; if the calculated Chi-square statistic is less than this value, you reject the null hypothesis.

Key Concepts

Left-Tailed Test

A left-tailed test is a hypothesis test where the critical region is in the left tail of the distribution. It is used when the alternative hypothesis states that the parameter is less than a certain value. The critical value marks the boundary below which we reject the null hypothesis.

Chi-Square Distribution for Variance/Standard Deviation Tests

When testing hypotheses about a population variance or standard deviation, the test statistic follows a chi-square distribution. This distribution depends on the sample size and degrees of freedom, which is n - 1. The shape is skewed, and critical values are found using chi-square tables.

Significance Level (α) and Critical Value

The significance level α represents the probability of rejecting the null hypothesis when it is true (Type I error). For a left-tailed test, the critical value corresponds to the chi-square value at the α percentile. This value determines the cutoff point for deciding whether to reject the null hypothesis.