In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.
c = 0.95, s^2 = 11.56, n = 30

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with constructing confidence intervals for (a) the population variance and (b) the population standard deviation, given a confidence level (c = 0.95), sample variance (s² = 11.56), and sample size (n = 30). The population is assumed to be normally distributed, which allows us to use the Chi-Square distribution for this calculation.
2. Step 2: Identify the formula for the confidence interval of the population variance. The confidence interval for the population variance (σ²) is given by: \( \left( \frac{(n-1)s^2}{\chi^2_{\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}} \right) \), where \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) are the critical values of the Chi-Square distribution with \( n-1 \) degrees of freedom.
3. Step 3: Calculate the degrees of freedom and the critical values. The degrees of freedom (df) are \( n-1 \), so \( df = 30-1 = 29 \). Using a Chi-Square table or calculator, find the critical values \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) for a 95% confidence level. Here, \( \alpha = 1 - c = 0.05 \), so \( \alpha/2 = 0.025 \).
4. Step 4: Plug the values into the formula for the confidence interval of the population variance. Substitute \( n-1 = 29 \), \( s^2 = 11.56 \), and the critical values \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) into the formula to compute the lower and upper bounds of the confidence interval for the population variance.
5. Step 5: To find the confidence interval for the population standard deviation (σ), take the square root of the lower and upper bounds of the confidence interval for the population variance. This will give you the confidence interval for σ.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population parameter.

Chi-Squared Distribution

The Chi-squared distribution is a statistical distribution that is used to estimate the variance of a population based on sample data. It is particularly important when constructing confidence intervals for population variance and standard deviation, as it allows us to determine the critical values needed for these calculations, especially when the sample size is small.

Sample Variance and Standard Deviation

Sample variance (s²) is a measure of how much the values in a sample differ from the sample mean, while the standard deviation (s) is the square root of the variance. These statistics are crucial for estimating the population variance and standard deviation, as they provide the necessary information to calculate confidence intervals and assess the variability within the data.