In Exercises 9–12, construct the indicated confidence interval...

Question

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

c = 0.95, s^2 = 11.56, n = 30

Accepted Answer

Welcome back, everyone. In this problem, a quality control team collects a random sample of 30 batteries and finds that the sample variance in battery life is 11.56 squared hours. Assume the data are from a normally distributed population. Construct the 95% confidence interval for the population variance and the population standard deviation. Now, for us to do that, let's first make note of the information that we have, OK? So far, we are given a confidence level C of 95% or 0.95%, a sample size N of 30 for 30 batteries. And a sample variance is squared of 11.56 squared over. Now, let's start by trying to figure out the, sorry, the 95% confidence interval for the population variance, OK? Now, how can we do that? Well, recall, OK, that for a normally distributed population, the confidence interval for the population variance, OK, the confidence interval for the population variance. Would be equal to. N minus 1, OK, multiplied by the variance, divided by our right chi squared critical value. So, let me just make sure I draw that symbol properly. So our right chi squared critical value. And The upper bound of our interval would be equal to N minus 1 divided by or sorry, multiplied by our variance divided by the left critical chi squared value. So in this, in both of these cases, we need to figure out our chi squared values from the right, or sorry, our right and left chi squared critical values from the chi squared distribution. No, We also know here. Then, since, well, or rather, since we have N and or variance, let's work on those values. So how can we find that? Well, we can go to the chi square distribution table and know from our table at a confidence level of 95%. And for the degrees of freedom, which equals N minus 1, 2130 minus 1, which is 29. So for a confidence level of 95% and 29 degrees of freedom. Then our critical high squad value from the left, OK. OK, it's gonna be our core value. For the 0.0 to 5th percentile, which is 45.722. OK. And our critical k square value from the right is gonna be our chi squad value at the 97.5th percentile, which is 16.047. Know that we have both of those, then we can find our limits for our confidence interval because no, this implies then that the limits for the uh population variance, OK, is gonna be equal to N minus 1, we know that's 29 multiplied by our variance of 11.56. Divided by our chi square value on the left, critical chi square value 4 is 5.722. That's for a lower limit, while for the upper limit, it's gonna be 29 multiplied by 11.56 divided by 16.047. OK No, when we go ahead and solve here. Then we should get this interval to go from 7.33. On the left, they are from our lower limit. To 20.89 for our upper limit, approximately. So this is our 95% confidence interval for the population variance. Now, remember, the next part of our problem we wanted to solve for the confidence interval for the population standard deviation, OK? But. Population standard deviation. And now to do that, remember that the standard deviation is the square root of the variance. So all we need to do is to take the square root of both of our end points. So that's gonna be the square root of 7.33 and the square root of 20.89, which is gonna be approximately equal. To 2.71 for a lower limit and 4.57 for our upper limit. Therefore, the 95% confidence interval for or variance is 7.33 to 20.89 and for our standard deviation, it's 2.71 to 4.57. Thanks a lot for watching everyone. I hope this video helped.

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

Key Concepts

Confidence Interval

Chi-Squared Distribution

Sample Variance and Standard Deviation

Watch next

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

Key Concepts

Confidence Interval

Chi-Squared Distribution

Sample Variance and Standard Deviation

Watch next

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