a. Determine the critical value for a right-tailed test of a population standard deviation with 18 degrees of freedom 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 this is a right-tailed test for a population standard deviation, we use the Chi-square (\( \chi^2 \)) distribution.
2. Determine the degrees of freedom (df): For a sample size \( n \), the degrees of freedom for variance or standard deviation tests is \( df = n - 1 \). Here, \( df = 18 \).
3. Identify the significance level \( \alpha \): Given \( \alpha = 0.05 \) for a right-tailed test, this means the critical value corresponds to the upper 5% of the Chi-square distribution with 18 degrees of freedom.
4. Use a Chi-square distribution table or statistical software to find the critical value \( \chi^2_{\alpha, df} \) such that the area to the right of this value is 0.05 with 18 degrees of freedom.
5. The critical value you find will be the cutoff point where if the test statistic exceeds this value, you reject the null hypothesis in favor of the alternative hypothesis.

Key Concepts

Chi-Square Distribution

The chi-square distribution is used in hypothesis testing for population variances and standard deviations. It is a right-skewed distribution that depends on degrees of freedom, which are related to sample size. For a given significance level, critical values are found from this distribution.

Degrees of Freedom

Degrees of freedom (df) represent the number of independent values that can vary in a calculation. In variance or standard deviation tests, df is typically the sample size minus one. It determines the shape of the chi-square distribution and is essential for finding the correct critical value.

Significance Level and Critical Value

The significance level (α) is the probability of rejecting the null hypothesis when it is true. For a right-tailed test, the critical value is the chi-square value where the area to the right equals α. This value sets the threshold for deciding whether to reject the null hypothesis.