Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.
c = 0.90, n = 8

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Determine the degrees of freedom (df) using the formula: df = n - 1, where n is the sample size. For this problem, df = 8 - 1.

Identify the level of confidence (c) and calculate the significance level (α) using the formula: α = 1 - c. For c = 0.90, α = 1 - 0.90.

Divide the significance level (α) into two tails for a two-tailed test. The left tail will have α/2, and the right tail will have α/2.

Use a chi-square distribution table or statistical software to find the critical values χL² and χR². For the left critical value (χL²), use df and the cumulative probability of α/2. For the right critical value (χR²), use df and the cumulative probability of 1 - α/2.

Verify the critical values by ensuring they correspond to the correct cumulative probabilities in the chi-square distribution table or software output.

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Chi-Square Distribution

The Chi-Square distribution is a statistical distribution that is used primarily in hypothesis testing and confidence interval estimation for categorical data. It is defined by its degrees of freedom, which are determined by the sample size and the number of categories. The distribution is positively skewed, meaning it has a longer tail on the right side, and is used to assess how observed data compares to expected data.

Critical Values

Critical values are the threshold points that define the boundaries of the acceptance region in hypothesis testing. They are determined based on the desired level of confidence (c) and the degrees of freedom associated with the test. For a Chi-Square test, critical values help to determine whether to reject the null hypothesis by comparing the test statistic to these values.

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of the Chi-Square distribution, degrees of freedom are typically calculated as the number of categories minus one or as the sample size minus one. They play a crucial role in determining the shape of the Chi-Square distribution and the corresponding critical values.

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.c = 0.90, n = 8