Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.
Car Batteries The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

Accepted Answer

Step 1: Identify the given information. The sample size (n) is 18, the sample standard deviation (s) is 0.25 hours, and the confidence level is 80%. The goal is to construct a confidence interval for the population variance (σ²). Step 2: Recall the formula for the confidence interval of the population variance. The confidence interval is given by: $$ \left( \frac{(n-1)s^2}{\chi^2_{	ext{right}}}, \frac{(n-1)s^2}{\chi^2_{	ext{left}}} ight) $$, where $$ \chi^2_{	ext{right}} $$ and $$ \chi^2_{	ext{left}} $$ are the critical values of the chi-square distribution for the given confidence level. Step 3: Calculate the degrees of freedom (df). The degrees of freedom is $$ n-1 $$, so $$ df = 18 - 1 = 17 . $$Step 4: Determine the critical values $$ \chi^2_{	ext{right}} $$ and $$ \chi^2_{	ext{left}} $$ from the chi-square distribution table for $$ df = 17 $$ and an 80% confidence level. The confidence level splits the remaining 20% into two tails, so the left tail has 10% (0.10) and the right tail has 10% (0.90). Step 5: Plug the values into the confidence interval formula. Use $$ s^2 = (0.25)^2 $$ and the critical values $$ \chi^2_{	ext{right}} $$ and $$ \chi^2_{	ext{left}}  to $$compute the lower and upper bounds of the confidence interval for the population variance. Finally, interpret the interval in the context of the problem.

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Key Concepts

Confidence Interval

Population Variance

Sample Standard Deviation