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 = 0.99, s^2 = 0.64, n = 7

c = 0.99, s^2 = 0.64, n = 7

Accepted Answer

Hello everyone. Let's take a look at this question together. A study analyzed the variation in reaction times in seconds squared for a sample of seven individuals. The sample variance was found to be 0.64. Assume the data come from a normally distributed population. Construct the 99% confidence interval for the population variance and the population standard deviation. So in order to solve this question, we have to recall how to construct a confidence interval so that we can construct a 99% confidence interval for the population variance and the population standard deviation of the reaction times in seconds squared for a sample of seven individuals where. The sample variance was found to be 0.64, and we must also assume that the data comes from a normally distributed population. And so the first step in solving this problem is to identify the given information, which we know that we are given the following, which is a confidence level of 0.99. A sample size N of 7 and a sample variance which is our Squad value is equal to 0.64, and then we have to use the formula to calculate our confidence interval, which for a normally distributed population, the confidence interval or the following, which is the population variance, is given using the following formula for the population variance and. Then for the population standard deviation, we have the square root of the population variance formula, where in these formulas, we know that our chi square sub-R and chi square subL are the right and left critical values from the chi square distribution, and we also know that our degrees of freedom, which is calculated as N minus 1, is equal to 6. And then we can determine our Critical chi square values, which at our confidence level of 0.99, the left and right critical values for a degree of freedom equaling 6 are given as the following. Using the chi square distribution table, we know that our left critical value is equal to chi square 0.005, which is equal to 18.548, and our right critical value is given as. square 0.995, which is equal to 0.676, and then we substitute the values to calculate our confidence interval for the population variance, starting with the lower limit, which we know is equal to 6 multiplied by 0.64 divided by 18.548, giving us our lower limit value of approximately 0.21. Then we calculate for the upper limit, which is also 6 multiplied by 0.64, but divided by 0.676, giving us our upper limit value of approximately 5.68. And then we substitute those values to calculate our confidence interval for the standard deviation confidence interval, which we know that for the lower limit and the upper limit are the square. Roots of our confidence interval for the variance, so we plug in the values that we used for the lower limit in the confidence interval for variants and take the square root of those values, giving us our lower limit of approximately 0.46, and then we calculate the same for the upper limit, so our upper limit is approximately 2.38. Therefore, we have our confidence interval. For the variant, which is 0.21 to 5.68, and our confidence interval for the standard deviation, which is 0.46 to 2.38. And looking at our answer choices, we can identify that this is answer choice C, so answer choice C is the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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 = 0.99, s^2 = 0.64, n = 7

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 = 0.99, s^2 = 0.64, n = 7

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 = 0.99, s^2 = 0.64, n = 7