Arm Circumferences Listed below are arm circumferences (cm) of...

Question

Arm Circumferences Listed below are arm circumferences (cm) of randomly selected women (based on Data Set 1 “Body Data” from Appendix B). Also shown is the normal quantile plot of those measurements.

b. Are the requirements for constructing a 95% confidence interval estimate of the population standard deviation satisfied? If so, construct that confidence interval.

Accepted Answer

All right, hello everyone. So this question says 12 batteries are randomly selected and tested for their life spans in hours, producing the following values. The normal quantile plot of this data appears to follow a straight line. Are the necessary assumptions satisfied to construct a 95% confidence interval for the population standard deviation? If so, calculate the interval. And here we have 4 different answer choices labeled A through D. So here our first step is to calculate the mean of the sample or X-bar. Now, recall that X bar is equal to the sum of all values in the sample, divided by the number of values there are. In this case, the sample size n is equal to 12. So to calculate X bar, you would add together all the data points given and divide that sum by 12. This ultimately equals 50, or excuse me, 514.9. Which divided by 12, gives you the sample mean of 42.91. And so if I scroll down here to open up some space, you can see that I went ahead and pre-wrote a table. To calculate the standard deviation for each of the data points given. Now, standard deviation, if you recall. First requires that you take each individual value, subtract that from the mean, and then square the difference. So taking our first data point, for example, that's 42.1, you would subtract this from the mean, that's 42.91, and square it, and this would give you 0. 656 1. And here on the side, I'm just illustrating that calculation as a reference for the other data points. So now for 43.7. That would give you 0.6241. 441.5, that's 1.9881. For 44.2, that's 1.6641. For 42.8. That's 0. 0121 443.3. That's 0.1521. For 41.9, that's 1.0201. For 44.0, that's 1.1881. 42.5, that would be 0.1681. 43.1 gives you 0.0361. 42.3 gives you 0.3721. And 43.5 is 0.3481. So just to go ahead and recap what we talked about earlier, what we've calculated at this point is the difference between each data point. Subtracted by the mean and squared. And so when you take the sum of all of those values. That gives you 8 point. Oops. 8.2292. That is adding together all of the values we just calculated gives you a total of 8.2292. Because it's the overall sum that's going to be relevant for the standard deviation. So sigma Is equal to the square root. Of the sum Of each data point subtracted by the mean squared. Divided by and subtracted by one. So that's the square root. Of 8.2292. Divided by 12, subtracted by 1. Which ultimately equals 0.865? So from here, we have to find the chi square critical values. First, the degrees of freedom. is equal to the sample size subtracted by 1, which in this case equals 11. Now, the confidence interval is 95%, which means that alpha is equal to 1 subtracted by 0.95 or 0.05. With this, we can find the left critical value. And the right critical value. The less critical value. Is X of 2. And alpha divided by 2. So that's 1 of 2. And 0.05 divided by 2, which equals 0.025. And this value corresponds to 21.92. The right critical value. Is X of 2 and 1 subtracted by alpha divided by 2. So that's 1 of 2. And one subtracted by 0.025, which equals 0.975. And so the right critical value is 3.82. And from here we can go ahead and calculate the upper and lower bounds of the 95% confidence interval. So the lower bound Is equal to the square root of. Let me actually make this bigger. The lower bound, as I said before, is equal to the square root of N subtracted by 1, multiplied by the standard deviation squared. Divided by The lower critical value or left critical value. The upper bound on the other hand, is equal to the square root of. And subtracted by one multiplied by the standard deviation squared. And divided by the right critical value. So starting off with the calculations for the lower bound. That's going to be equal to. 11, which is 12 subtracted by 1, multiplied by 0.865 squared. Divided by the left critical value of 21.92. And this gives you the lower bound of the confidence interval equal to 0.61. The upper bound on the other hand. Is equal to 11. Multiplied by 0.865 squared. Divided by 3.82. And this equals 1.47. As the upper bound of the confidence interval. And so there you have it. So scrolling up here to consider the answer choices. Our correct one is going to be option D. Since assumptions are satisfied and the 95% confidence interval is from 0.61 to 1.47. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Confidence Interval

Normal Distribution

Sample Size and Random Sampling

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