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.

Annual Precipitation The annual precipitation amounts (in inches) of a random sample of 61 years for Chicago, Illinois, have a sample standard deviation of 6.46. Use a 98% level of confidence. (Source: National Oceanic and Atmospheric Administration)

Accepted Answer

Hello everyone. Let's take a look at this question together. The weights in kilograms of 41 randomly selected boxes of sugar at a sample standard deviation of 1.8 kg. Assume the weights are normally distributed using a 98% confidence level, struct 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 a 98% confidence level of the weights in kilograms of 41 randomly selected boxes of sugar, which we know had a sample standard deviation of 1.8 kg, and we must also assume the weights are normally distributed. Then when we construct our confidence interval, we must also interpret the results. And so the first step in structing our confidence interval is to identify our given values, which we know that the information we are given is a sample size N of 41. The degrees of freedom we can calculate now is equal to N minus 1, so our degrees of freedom is equal to 40. Our sample standard deviation is 1.8. We note the confidence level is equal to 98%. And our alpha is equal to 0.02, therefore, an alpha divided by 2 is equal to 0.01. And now we can use the chi square table values, which we know from the chi square distribution table for degrees of freedom equaling 40. The right tail critical value is equal to 63.691, and the left tail critical value is equal to. 22.164. And then we can use the formula for confidence interval for population variance, which is given as the following formula, which from this formula, we must first compute our Squared value, which is equal to 1.8 squared, or 3.24, and then N minus 1 multiplied by Squared is equal to 40 multiplied. 3.24, so this is equal to 129.6. And so using the formula for the confidence interval for population variance, we plug in our values that we have computed, which gives us the lower limit, which is equal to 129.6 divided by 63.691, so our lower limit is approximately 2.0348. And then we do the same for the upper limit, which is 129.6, divided by 22.164, which is approximately 5.8473. Therefore, from the lower and upper limits, we know the confidence interval for the population variance is 2.0 to 5. 8. And interpreting these results, we know that this confidence interval tells us we are 98% confident that the true population variance in sugar box weights lies between 2.0 and 5.8 kg squared, and based on our calculated confidence interval and our interpretation. We can identify that answer choice C is the correct answer, as it contains the correct confidence interval for the population variance and the correct interpretation that we are 98% confident that the true population variance in sugar box weights lies between 2 and 5.8 kg squared. 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.

Key Concepts

Confidence Interval

Population Variance

Sample Standard Deviation

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