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.

Volleyball The numbers of service aces scored by 15 teams randomly selected from the top 50 NCAA Division I Women’s Volleyball teams for the 2021 season have a sample standard deviation of 26.1. Use an 80% level of confidence. (Source: National Collegiate Athletic Association)

Accepted Answer

Hello, everyone. Let's take a look at this question together. The number of defective light bulbs found in 10 randomly selected shipments had a sample standard deviation of 4.2 bulbs. Assume the number of defects follows a normal distribution. Using an 80% confidence level, construct a confidence interval for the population standard deviation. Also interpret the results. So in order to solve this question, we have to recall how to construct a confidence interval for the population standard deviation, so that we can construct one using an 80% confidence level, and we know the number of defective light bulbs found in 10 randomly selected shipments at a sample standard deviation of 4.2 volts. And we must also assume The number of defects follows a normal distribution. Then when we construct our confidence interval, we interpret the results. And so the first step in constructing our 80% confidence interval or population standard deviation is to identify our given values which we know that we are given the following, which is a sample standard deviation of 4.2. A sample size of 10, our degrees of freedom is calculated as N minus 1, so our degrees of freedom is equal to 9, our confidence level is 80%, and our alpha is equal to 1 minus 0.80, which is from the confidence level, which equals 0.20, so then our alpha divided by 2 is 0.10. And then we We can use the formula for confidence interval for population variance, which is given as the following formula, where from this formula we must first compute the value for our sample variants as well as N minus 1 multiplied by our sample variance, which we know that our sample standard deviation is 4.2, so the sample variance is 4.2 squared, which is equal to 17. 64 and then plugging in the values for and minus 1 multiplied by our sample variants. We have 9 multiplied by 17.64, which is equal to 158.76. And then from the chi score distribution table for our degrees of freedom, which is equal to 9, our right tail critical value is 14.684, and our Left tail critical value is 4.168, so our lower limit is calculated as 158.76 divided by 14.684, which is approximately 10.81, and our upper limit is equal to 158.76 divided by 4.168, which is approximately 38.09. So our lower limit is approximately 10.81, and our upper limit is approximately 38.09, which gives us the confidence interval for the variance of 10.8 to 38.1. However, we know we are looking for the confidence interval for the population standard deviation. So we have to take the square roots for the standard deviation confidence interval, and the square root of 10.81, and the square root of 38.09 is approximately 3.3 to 6.2. So the confidence interval for the population standard deviation is 3.3 to 6.2, and then interpreting our results, we know we are 80% confident that the true population standard deviation in defective light bulbs per shipment. It is between 3.3 and 6.2 bulbs. 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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