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 = 0.98, s^2 = 278.1, n =41

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with constructing confidence intervals for (a) the population variance (σ²) and (b) the population standard deviation (σ) using the given sample variance (s² = 278.1), sample size (n = 41), and confidence level (c = 0.98). 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 is given by: \( \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}} \), where \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) are the critical values of the Chi-Square distribution with \( df = n-1 \) degrees of freedom, and \( \alpha = 1 - c \).
3. Step 3: Calculate the degrees of freedom and the significance level. The degrees of freedom (df) is \( n-1 = 41-1 = 40 \). The significance level \( \alpha \) is \( 1 - c = 1 - 0.98 = 0.02 \). Divide \( \alpha \) by 2 to find \( \alpha/2 = 0.01 \).
4. Step 4: Look up the critical values of the Chi-Square distribution. Using a Chi-Square table or statistical software, find \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) for \( df = 40 \). These values correspond to the 0.01 and 0.99 percentiles of the Chi-Square distribution, respectively.
5. Step 5: Plug the values into the formula for the confidence interval of the population variance. Substitute \( n-1 = 40 \), \( s^2 = 278.1 \), and the critical values \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \) into the formula to calculate the lower and upper bounds of the confidence interval for \( \sigma^2 \). To find the confidence interval for the population standard deviation \( \sigma \), take the square root of the lower and upper bounds of the variance confidence interval.

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 98% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 98% 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 the test statistic follows a chi-squared distribution when the population is normally distributed.

Sample Variance and Standard Deviation

Sample variance (s²) measures the spread of sample data points around the sample mean, while the standard deviation (s) is the square root of the variance. These metrics are crucial for estimating the population variance and standard deviation, especially when constructing confidence intervals, as they provide insight into the variability of the data.