In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.


Left-tailed test, n=7,α=0.01

Verified step by step guidance

1. Step 1: Understand the problem. This is a chi-square test, specifically a left-tailed test. The goal is to find the critical value(s) and rejection region(s) based on the sample size (n=7) and the level of significance (α=0.01).
2. Step 2: Recall the formula for degrees of freedom in a chi-square test. Degrees of freedom (df) are calculated as df = n - 1, where n is the sample size. For this problem, df = 7 - 1 = 6.
3. Step 3: Use the chi-square distribution table or a statistical software to find the critical value for a left-tailed test with df = 6 and α = 0.01. In a left-tailed test, the critical value corresponds to the area to the left of the curve equal to α.
4. Step 4: Define the rejection region. For a left-tailed test, the rejection region is the range of chi-square values less than the critical value obtained in Step 3. This region represents where the null hypothesis would be rejected.
5. Step 5: Summarize the findings. The critical value and rejection region are determined based on the chi-square distribution table or software output. Ensure you understand how to interpret these values in the context of hypothesis testing.

Key Concepts

Chi-Square Test

The chi-square test is a statistical method used to determine if there is a significant association between categorical variables. It compares the observed frequencies in each category to the frequencies expected under the null hypothesis. This test is commonly used in hypothesis testing to assess goodness-of-fit or independence in contingency tables.

Critical Value

A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α) and the distribution of the test statistic. For a left-tailed chi-square test, the critical value indicates the point below which the test statistic must fall to reject the null hypothesis.

Rejection Region

The rejection region is the set of values for the test statistic that leads to the rejection of the null hypothesis. In a left-tailed test, this region is located to the left of the critical value on the chi-square distribution. If the calculated test statistic falls within this region, it suggests that the observed data is unlikely under the null hypothesis, prompting its rejection.