Testing the Difference Between Two Means (c) calculate d̄ and ...

Question

Testing the Difference Between Two Means (c) calculate d̄ and Sd, Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Migraines

A researcher claims that injections of onabotulinumtoxinA reduce the number of days per month that chronic migraine sufferers have headaches. The table shows the number of days chronic migraine sufferers suffered migraines before and after using the treatment. At , α= 0.01 is there enough evidence to support the researcher’s claim? (Adapted from Journal of Headache and Pain)

Table comparing the number of migraine days before and after treatment for chronic migraine patients.

Accepted Answer

Hi everybody. Our next problem says a coach records the number of sit-ups completed by 7 athletes before and after a 3-week training program, and then we're provided with those number of sit-ups for the 7 athletes, a list for before, a list for after. We'll come back to those numbers. And then it says calculate the mean, the bar, so the mean difference, and the standard deviation S of the differences before minus after. And our answer choices are A, D bar equals 3.8, S equals 1.1. B D equals -3.8 S equals 0.3. C D equals 3.1, S equals 0.5, or D D equals -3.1, and S equals 0.4. So, let's think about how we calculate those numbers. So for the mean, so Bard. We will need to look at. The sum of all of the differences. And then divide by the number of athletes. So first, let's look at our differences here. So, Our first set of samples, and remember we're doing before minus after, or before 30 after 33. So the difference is -3. And I'm going to write these in next to my bard to fill in my equation, because I'm going to add them all together, so that I can add. Now let's look at the next pair. Is 28 before 31 after, so that's also going to be. -3 is the difference. Next pair, 32 and 36. So this one has a difference of -4. Our next pair, 35 and 38, so another -3. Next pair 29 and 32. So another -3. Next pair 31 and 34. -3. And our final pair 27 and 30, so that is also -3. Now, notice that our main difference here is -3 in 123456 cases, and -4 and only 1. So, we're going to expect a very small standard deviation. So let's go ahead and divide this, and we divide it by the number of Uh, objects in the sample, or then that would be 7 athletes, so divide that all by 7. So that's going to end up equaling. 22, divided by 7, which will be approximately. -3.14. So that's going to be our main difference. So we can go ahead and look at our answer choices if we want, see if we can eliminate something. They're all expressed to just to the 10th place, so that would be approximately 3.2. So, Choice A says that uh Bar D or Db bar is 3.8, so that's incorrect. So let make choice A, Choice B also says uh D bar is -3.8. Choice C says D bar is 3.1, and choice D has D bar equaling 3.1. So, C is also incorrect, because this is a positive value. So actually, we don't even need to, if we were on a test, you wouldn't even need to go any farther. You'd know choice D was your answer, but this is a practice problem, we want to be thorough. Let's calculate our standard deviation. But you would be grateful on a test since the standard deviation is a longer calculation. So, remember, this is a sample standard deviation, not a population. So, that formula is the square root, and then we have a fraction under the square root. In the numerator, we have the sum of, and in parenthesis, X minus X bar, close parenthesis, and that's squared. And then underneath. N minus 1, all of that under the square root sign. Excuse me, N minus 1 in the denominator. So, basically, we sum up the squares of the differences from the mean for each of these. Well, we're made, things are made uh quite a bit simpler by the fact that 6 of our measurements have the same difference from the mean. So, in terms of difference from the mean, we have -3. And we would subtract 3.14 or subtract the D bar, which is negative 3.14, so we add 3.14. And we're going to square that. Some And that's going to equal 0.0. 196. And recall that we have that 6 times. So, multiply that number. By 6. So that makes it looks like it's equal, so, just say 6 times. And then our other value, -4 plus 3.14, square that, and that will equal 0.7396, and that's only one time. So, then we need to to add together, it's the sum of those squared differences. So, our standard deviation is going to equal the square root, and then in the numerator, we will have 6 multiplied by 0.0196. And then add 0.7396, and then in the denominator, N minus 1, and is our sample number, which is 7. 7 minus 1 and the denominator all that under the square root sign. So, I will give you a moment to put those numbers in your calculator. You can pause here if you wish. And you will get the answer 0.378 for our standard deviation for our sample. And our answer choices are given in just the 10th place, so that would be round off to 0.4, and that does correspond with what we have in choice D. So again, we're given Before and after numbers of sit-ups, before and after a three-week training program, and we're supposed to calculate the mean difference and the standard deviation for these values before minus after. Our answer is choice D. the mean difference is -3.1, and the standard deviation is 0.4. See you in the next video.

Key Concepts

Dependent Samples

Mean Difference (d̄)

Standard Deviation of Differences (Sd)

Watch next