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 = 0.80, n = 51

Verified step by step guidance

1. Determine the degrees of freedom (df) using the formula: df = n - 1, where n is the sample size. In this case, df = 51 - 1.
2. Identify the level of confidence (c) and calculate the significance level (α) using the formula: α = 1 - c. For c = 0.80, α = 1 - 0.80.
3. 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 also have α/2.
4. Use a chi-square distribution table or statistical software to find the critical values χL² and χR². For χL², find the value corresponding to the cumulative probability of α/2 with df degrees of freedom. For χR², find the value corresponding to the cumulative probability of 1 - α/2 with df degrees of freedom.
5. Verify the critical values by ensuring they correspond to the correct cumulative probabilities and degrees of freedom in the chi-square distribution table or software.

Key Concepts

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 parameters being estimated. The distribution is positively skewed, and as the degrees of freedom increase, it approaches a normal distribution.

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 specific distribution being used, such as the Chi-Square distribution. For a given confidence level, critical values help in deciding whether to reject or fail to reject the null hypothesis.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of the Chi-Square distribution, degrees of freedom are typically calculated as the sample size minus one (n - 1) for a single sample. They play a crucial role in determining the shape of the Chi-Square distribution and the corresponding critical values.