In Exercises 5–20, assume that the two samples are independent...

Question

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)

Magnet Treatment of Pain People spend around $5 billion annually for the purchase of magnets used to treat a wide variety of pains. Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results given below are among the results obtained in the study (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study,” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). Higher scores correspond to greater pain levels.

c. Does it appear that magnets are effective in treating back pain? Is it valid to argue that magnets might appear to be effective if the sample sizes are larger?

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Accepted Answer

Welcome back everyone. In this problem, a pharmaceutical company wants to determine if their new drug reduces cholesterol more effectively than a placebo. The reduction in cholesterol in milligrams per deciliter was measured after 3 months. The data are as follows. For the new drug group, the sample size is 22, the mean reduction is 15.6, and the standard deviation is 4.2. For the placebo group, the sample size is 22, mean reduction 14.9, and standard deviation is 5. Assume that the two samples are independent simple random samples selected from normally distributed populations and do not assume that the population standard deviations are equal. Use degrees of freedom equal to the smaller of N1 minus 1 and N2 minus 1. Does it appear that the new drug reduces cholesterol more effectively than the placebo? And is it valid to argue that the drug might show a more significant effect with larger sample sizes? Now let's focus on the first part of this problem. Does it appear that the new drug reduces cholesterol more effectively than the placebo? So we are testing on a hypothesis. So since we're testing, if the new drug leads to a greater mean reduction, then our null hypothesis. OK, first, or null hypothesis is that the new drug is not more effective, that is, mu1 is less than or equal to m2. Which means that the alternative hypothesis is that mu 1 is greater than mu 2. In other words, the new drug is more effective. Know from the information that we have, OK, for our first group. OK, being the new, the group that took the new drug, the sampleine XB 1 is 15.6. The sample standard deviation is 4.2 and the sample size is 22. On the other hand, for the placebo group, if we consider it to be the second group, then its sample mean is 14.9, sample standard deviation is 5, and its sample size is 22. Now to test our hypothesis, we'll need the test statistic and the critical T value. Because if the test statistic is greater than the critical value, OK, since this is gonna be a one-tailed test. If the test statistic is greater than the critical value and falls in the region shaded in red, then there is sufficient evidence to reject the null hypothesis. But if not, then we cannot reject the null hypothesis. So we'll need both of those values to compare. Let's start by trying to find the test statistic. We can use the formula for the two sample T test of unequal variances. Now, that tells us the T value will be equal to the difference between the sample means, OK. Minus the difference between the population means which it's often assumed to be zero. Divided by the square root, OK, of S1 squared divided by N1 plus S2 squared divided by N2. If we go ahead and substitute the values, that's going to be 15.6 minus 14.9, OK, minus 0. All divided By the square root of 4.2 square divided by 22 plus 5 square divided by 22. Think I can even make some space here, OK. So we know that if it subtracts 0 ended. Numerator, it will leave us with the expression that was there already. So no, when we go ahead and solve, then we should find that the test statistic is approximately 0.503. Next, we need the critical values. So for this test, OK, it's a one tailed test. At a significant level alpha of 0.05 and degrees of freedom that's one less than the smaller of both sample sizes. Since both sample sizes equal 22, then we use 21 degrees of freedom. And for this one tail test under these conditions, then we should get the critical value to be approximately 1.721. So what does that mean for us? Well, no, since our test statistic is equal to 0.503, which is less than our critical value of 1.7 to 1, then we fail to reject the null hypothesis. So what does that mean for us? Well, if we can write our statement as a conclusion here, OK, then that means we can say that at the 0.05 significance level. OK. There is not enough evidence to support the claim. OK That the new drug. Leads to a greater mean reduction in cholesterol. OK. And the placebo, because our our our test statistic is less than the critical value. Now what about the second part of our problem? Remember, the second part asks us to figure out if it's valid to argue that the drug might show a more significant effect with larger sample sizes. So in effect, we want to know what does a larger sample size do to our our study. Well, with larger sample sizes, the standard error decreases, and that could make it easier to detect a significant difference if one exists. Therefore, the answer is yes. So we can say that at the 0.05 significance level, there isn't enough evidence to support the claim that the new drug reduces cholesterol more effectively than the placebo and a larger sample size might show a more significant effect. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Independent Samples

Visual Analog Scale (VAS)

Hypothesis Testing

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