Constructing Confidence Intervals In Exercises 13–24, assume t...

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

Hello, everyone. Let's take a look at this question together. The durability times in hours of 12 randomly selected ceiling fans have a sample standard deviation of 0.4 hours. Assume the data come from a normally distributed population. Using an 80% confidence level, construct the confidence interval for the population variance. Also interpret the results. So in order to solve this question, we have to recall how to construct a confidence interval for the population variance, so that we can construct one using an 80% confidence level for the durability times in hours of 12 randomly selected ceiling fans, which have a sample standard deviation of 0.4 hours, and we assume that the data comes from a normally distributed population. Then when we construct our confidence interval, we must also interpret the results. And so the first step in constructing our confidence interval is to identify the given values which we know that from the provided information we are given the following values, which are a sample standard deviation of 0.4 hours. A sample size of 12. And from the sample size, we calculate the degrees of freedom, which is equal to and minus 1, so our degrees of freedom is equal to 11. Our confidence level is 80%. And then our alpha is equal to 1 minus 0.80, which comes from our confidence level of 80%, so 1 minus 0.80 is 0.20. Therefore, an alpha divided by 2 is equal to 0.10. And then we use the formula for confidence interval for population variance, which is the following formula, and then we calculate the values to substitute into that formula. which includes our sample variants that we calculate by squaring the sample standard deviation. So we have 0.4 squared, which is equal to 0.16, so our sample variance is equal to 0.16, and then plugging that into a minus 1 multiplied by our sample variance, we have 11 multiplied by 0.16, which equals 1.76. And then we need to Get our chi square critical values from a chi square distribution table, which from the chi square distribution table or our degrees of freedom, which is equal to 11, our right tail critical value is 17.275, and our left tail critical value is 5.578. And then we can calculate. Our lower limit and our upper limit values starting with the lower limit, which is equal to 1.76 divided by 17.275, which is approximately 0.1, and then for our upper limit, which is equal to 1.76 divided by 5.578, is approximately 0.3. So our lower limit value is approximately 0.1, and our upper limit value is approximately 0.3. Therefore, our confidence interval or variance is 0.1 to 0.3, and based on these results, we can interpret that we We are 80% confident that the true population variance in ceiling fan durability times is between 0.1 and 0.3 hours squared. And so looking at our answer choices, we can identify that answer choice B is the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Confidence Interval

Population Variance

Sample Standard Deviation

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