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.

Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

Table displaying drug concentration peak times in minutes for 15 patients, with values ranging from 2.92 to 19.08.

Accepted Answer

Hello everyone. Let's take a look at this question together. The drying times in minutes of a new eco-friendly paint were recorded for a sample of 15 wooden boards. Assume the drying times are normally distributed. The recorded times are the following data values. Construct a 90% 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 so that we can construct a 90% confidence interval for the population variance of the recorded drying times in minutes of a new eco-friendly paint, which were recorded for a sample of 15 wooden boards, and we assume the drying times are normally distributed, then we need To interpret our results, and the first step in solving this problem is to organize our data and compute, which we will first compute the sample mean and the sample variants using the following formula. More specifically, we will use both of the following formulas for the sample mean and the sample variants, where the sample mean is calculated as the sum of our. Values divided by our sample size N, and our sample variance is calculated as the sum of all of our X values subtracted by the sample mean squared divided by our sample size N minus 1. And from the data that we are provided, we can calculate the sum of our X values divided by N for the sample mean, which the sum of our X values. is equal to 450. So then our sample mean is calculated as 450 divided by 15, which is equal to 30. And then for the sample variants, we can take our X values subtracted by the sample mean, which is equal to 30, and square the value and then take the sum of all of those values to calculate the sample variants, which we know the sum of our X values subtract. Divide the sample mean squared is equal to 70, so then our sample variance is equal to 70, subtracted by 15 minus 1, which is equal to 5. So we have the value for our sample mean, which is 30, and our sample variance, which is 5. And now we can determine the degrees of freedom and get our critical values, which we know we have a sample size of 15. So our degrees of freedom, which is calculated as N minus 1, is equal to 14, and we know we have a confidence level of 0.90, so our alpha is equal to 1 minus 0.90, which is equal to 0.10, so then an alpha divided by 2 is equal to 0.05. And then we can determine the critical values, which from the chi square table for a degree of freedom. Which equals 14. Our right critical value, which is chi square 0.05, is equal to 23.685, and our left critical value, which is chi square 0.95, is equal to 6.571. And then using the following confidence interval formula for variants, we can substitute the values into the formula, which gives us 14 multiplied by 5 divided. By 23.685 to 14 multiplied by 5 divided by 6.571. And when simplified, our confidence interval or variance is 2.96 to 10.65. Therefore, the 90% confidence interval for the population variance is 3 to 10.7, and interpreting these results, we know we are 90% confident that the true population variances of drying times is between 3 and 10.7 minutes squared. So we have our 90% confidence interval for the population variance and our interpretation of the results, and looking at our answer choices, we can identify that the correct answer choice is answer choice B, 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 (σ²)

Normal Distribution

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