In Problems 5–8, find the critical values χ²₁₋ᵅ⁄₂ and χ²ᵅ⁄₂ for the given level of confidence and sample size.
6. 95% confidence, n = 25

Verified step by step guidance

1. Identify the degrees of freedom (df) for the chi-square distribution, which is given by \( df = n - 1 \). For this problem, calculate \( df = 25 - 1 \).
2. Determine the significance level \( \alpha \) from the confidence level. Since the confidence level is 95%, \( \alpha = 1 - 0.95 = 0.05 \).
3. Calculate the two tail probabilities for the chi-square distribution: \( \frac{\alpha}{2} = 0.025 \) and \( 1 - \frac{\alpha}{2} = 0.975 \). These correspond to the lower and upper critical values respectively.
4. Use a chi-square distribution table or statistical software to find the critical values \( \chi^2_{1 - \frac{\alpha}{2}} \) and \( \chi^2_{\frac{\alpha}{2}} \) for the degrees of freedom calculated in step 1 and the tail probabilities from step 3.
5. Interpret these critical values as the boundaries for the chi-square confidence interval for the variance or standard deviation, where \( \chi^2_{1 - \frac{\alpha}{2}} \) is the upper critical value and \( \chi^2_{\frac{\alpha}{2}} \) is the lower critical value.

Key Concepts

Chi-Square Distribution

The chi-square distribution is a probability distribution used primarily in hypothesis testing and confidence interval estimation for variance. It is defined by degrees of freedom, which typically relate to sample size minus one. The distribution is skewed right and used to find critical values for tests involving variance or categorical data.

Critical Values in Chi-Square Tests

Critical values are points on the chi-square distribution that correspond to specific probabilities, defining rejection regions for hypothesis tests or bounds for confidence intervals. For a two-tailed test, two critical values are found at α/2 and 1 - α/2, which mark the cutoff points for the confidence level.

Confidence Interval for Variance

A confidence interval for variance uses the chi-square distribution to estimate the range within which the true population variance lies with a given confidence level. The interval is calculated using the sample variance, degrees of freedom (n-1), and the chi-square critical values corresponding to the confidence level.