In Exercises 21–24, find the coefficient of variation for each...

Question

In Exercises 21–24, find the coefficient of variation for each of the two samples; then compare the variation. (The same data were used in Section 3-1.)

Pulse Rates Listed below are pulse rates (beats per minute) from samples of adult males and females (from Data Set 1 “Body Data” in Appendix B). Does there appear to be a difference?

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Accepted Answer

Welcome back, everyone. In this problem, we want to find a coefficient of variation for each sample and answer the following question. Listed below are systolic blood pressure readings in millimeters of mercury from samples of adult males and females. Does there appear to be a difference in variation between the two groups? For both of our samples, the male systolic blood pressure in millimeters of mercury is 120, 125, 130, 135, and 128, while the female systolic blood pressure is 110, 115, 118, 122, and 117. For our answer choices, A says male systolic blood pressure has a slightly higher relative variation than female systolic blood pressure. B says female systolic blood pressure has significantly higher variation compared to male systolic blood pressure. C says both male and female groups have the same level of variation in systolic blood pressure, and the D says the coefficient of variation suggests that male and female systolic blood pressures are not comparable. Now what are we going to do to help us find the coefficient of variation for each sample? What do we know about that? Well, recall, OK, that the coefficient of variation is equal to the standard deviation S divided by the mean for the respective group X bar multiplied by 100%. OK, where. The square of the standard deviation, the variance is equal to the sum of the difference between each data point and the mean squared, OK, divided by the sample size N minus 1. So, in order for us to find a coefficient of variation, first we'll need to find our variance, use that to find the standard deviation, and then divide it by the mean for each sample or each group. Now, let's go ahead then and do that. So to make this a bit easier, we're going to write or we're going to make two tables, OK? First, we're gonna have one for the males, and then we're gonna have one for the females, OK? So let me separate that here with this line. And then we know that for the meals. In our table, in order to help us to find the variances, we're going to have our, or we're gonna have each data point, OK. Have each data point, and then we're gonna have the difference between each data point and the mean, OK, Xi minus X bar, and then we're going to square that difference, OK? And we're going to do that for each data point. And essentially, we're going to do the same for the females, right? Once we have all of these samples or all of these squared values, OK. So we know for the males, it's 120, 125, 130, 135, and 128. When we have those values, OK, then we can use that to help us find the variants for the males, OK? And thus we'll be able to solve for the standard deviation and then the coefficient of variation within each group, OK? So let's go ahead and first populate our table. Realize that we'll need our mean. So what's the mean for the male group? Well, we know all of our male values here, and we know that there are 5 values, so the mean is gonna be equal to 120 plus 125 plus 130 plus 135 plus 128. That will all be divided by 5. And when we go ahead and solve here, we get the mean reading for me to be 127.6, OK? So now, for example. In our first value, for first data 0.120 the difference between it and the mean is gonna be 120 minus 127.6, which equals -7.6, OK? And no, that means when we square negative 7.6, it's going to be 57.76. Essentially, we're just going to repeat the same process for the rest of the data points. So for 125, we have 125 minus 127.6, which equals -2.6, and when we square that, it's 6.76. 130 minus 127.6 is 2.4, and it's square is 5.76. For 135, it would be, the difference would be 7.4. And thus its square would be 54.76. OK. And then for 128, the difference is 0.4 and that square is 0.16. Know that we have all the values, we are going to sum them, OK, so that we put them into our variance formula. So now when we add all of these values, OK, then we should get the sum of the squares of the differences to be equal to 100. Let me make some space actually. Just to show you what we're doing. We should get our total. To be equal to 125.2. So the variance will be 125.2 divided by our sample size 5 minus 1. And this would be equal to 3. 31.3. And if the variance is 31.3, that means the standard deviation is the square root of 31.3, which equals 5.59 or approximately 5.6 millimeters of mercury. Now that we know the standard deviation. This implies then that a coefficient of variation for males. OK. It's gonna be equal to 5.6 or standard deviation divided by the mean 127.6 multiplied by 100%, and this would give us a coefficient of variance of 4.39%. So now we know what the coefficient of variance is for males. No, we need to figure out what it is for females so that we can compare. So let me put females up here. And again to start, let's find the. Mean for our female data set. So now, that's gonna be equal to 110 plus 115 plus 118 plus 122 + 117, all divided by 5. That is equal to 116.4. So now if we make our table XI. X I minus X bar and X1 minus X bar squared, OK. And we populate it with our data points, OK. Let me just make a line here. Uh, I mean, I think I can make a better line. This is pretty good. So now we have uh like like we said, 110, 115, 118, 122, and 117, OK? No, When we do that here. For 110, the difference between it and the mean 116.4 is -6.4, and thus the square of that is 40.96. For 115, it's -1.4 and 1.96. For 118, it's 1.6 and 2.56. 122. 5.6 and 31.36. And for 117, 0.6 and 0.36. Now when we find a total here, when we sum that it's gonna be equal to 77.2. Now that we have that, OK, then that means the variance for the females is gonna be equal to 77.2 divided by 5 minus 1, and that would be equal to 19.3. This implies then that the that the uh standard deviation. For the female group is going to be equal to the square root of 19.3, which equals 4.39 millimeters of mercury. I know that we have that That means the coefficient of variation for females then is gonna be equal to 4.39 divided by 116.4, multiplied by 100%, which equals 3.77%. So the coefficient of variation for males is 4.39% while for females is 3.77%. Now we've done a lot of computation here, but what does this mean? Well, since the male systolic blood pressure has a CV of 4.39%, it indicates a slightly higher relative variability than the female systolic blood pressure of 3.77%. That indicates a lower relative variability. Thus we can conclude that the male blood pressure readings have slightly more variation relative to their mean compared to the female group. If we look back at our answer choices, that means a. Is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Coefficient of Variation

Standard Deviation

Mean

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