The data set represents the weights (in pounds) of 10 randomly...

Question

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

Table displaying weights in pounds of 10 randomly selected black bears from northeast Pennsylvania.

c. Construct a 99% confidence interval for the population standard deviation. Interpret the results.

Accepted Answer

All right, hello, everyone. So this question says, the following are the cholesterol levels in milligrams per deciliter of 10 randomly selected adults from a health survey. Assume the cholesterol levels are normally distributed. Construct a 99% confidence interval for the population standard deviation. And here we have not only the data set, but also 4 different answer choices labeled A through D. So, for this question, To construct a 99% confidence interval for the population standard deviation when the data is assumed to be normally distributed. We use the chi square distribution. So Notice how our sample size N is equal to 10. And that's going to be important for finding the sample variants. Now, to find the sample variants, you first have to know the sample mean. So the sample mean, or X bar, Is equal to The sum of all data points divided by the sample size. Adding up all of the data points gives you a total of 2140. So X bar is equal to 2140 divided by 10. Which gives you 214. And now to find the sample variants. But there's some information that you need to collect before you can do that, which is why I've created a table on the left side of the screen here. Because on the last column of this table, I've gone ahead and copied each of the data points given to us. And the reason I did that is because for each data point, the following calculation has to be made. So let's take our first data point, for example, of 108. The first step Is to Subtract the data 0.180 from the meat. Or from the sample meat that's 214. This eventually gives you a difference, which for 180 is -34. And now the next step is to square. The difference between X and X bar. So, 4 180, X subtracted by X bar squared is equal to 1,156. And so now you'd repeat this for all of the data points. So for 190, that's 576. For 195, that's 361. For 200, that's 196. For 210 that's 16. 4 to 20, that's 36. 4225, that's 121. For 230, that's 256. 4240, that's 676. And for 250, that's 1,296. The sum of. All values of X subtracted by X bar squared comes out to 4690. Which brings us now to the variants. Recall that the variance or as squared. Is equal to the sum of. All values of X subtracted by x bar squared. Divided by and subtracted by 1. So using the information found earlier. S squared Is equal to. 4690 Divided by 10, subtracted by 1. So, recall that the standard deviation S is simply equal to the square root of the variance. So is equal to the square root of 4690 divided by 9. And this Gives you 22 82. 8 And that's rounding to 3 significant, excuse me, 3 decimal places. So now, using the chi square distribution, our task is to find the confidence interval. So what is the formula for the confidence interval of the population variance sigma squared? It is as follows. So the confidence interval The lower bound that is of the confidence interval. Would it be and subtracted by one. Multiplied by squared. Divided by chi square of alpha divided by 2. The upper bound of the confidence interval. Is N subtracted by 1 multiplied by Squared. Divided by I square of 1 subtracted by alpha divided by 2. So let's take a moment to find the information that we have. Right, we know that the confidence level is 99%, which means that alpha is equal to 0.01. Which also means that alpha divided by 2 is equal to 0.005. The degrees of freedom, which are equal to and subtracted by 1, is equal to 9. So the chi square value. Of 0.005 with 9 degrees of freedom. Is equal to 23.589. And chi square of 0.995. With 9 degrees of freedom is equal to 1.735. Keep in mind that 0.995 comes from subtracting 0.005 from 1. So now let's find the confidence interval. Starting off with the lower limit. So the lower limit of the confidence interval is equal to 9, multiplied by 4690 divided by 9. And this in turn is divided by 23.589. On the other hand, at the upper limit. is equal to 9, multiplied by 4690 divided by 9. And divided by 1.735. The lower limit simplifies to 4690. Divided by 23.589. And the upper limit simplifies to 4690. Divided by 1.735. So, to get the interval for the standard deviation, you would take the square roots of both the lower and upper limits that you just found. So the square root Of the lower limit. is equal to Approximately 14.10. Whereas the square root of the upper limit. is equal to approximately 51.99. And so Your final answer for the confidence interval is from 14.10 to 51.99 mg per deciliter. And if I scroll up here to view the answer choices. This corresponds to option B. And there you have it So with that being said, thank you so very much for watching, and I hope you found this helpful.

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

c. Construct a 99% confidence interval for the population standard deviation. Interpret the results.

Key Concepts

Normal Distribution

Confidence Interval

Chi-Square Distribution

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