Performing a Sign Test In Exercises 7–22, (d) decide whether t...

Question

Performing a Sign Test In Exercises 7–22, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

[APPLET] Lower Back Pain A physician claims that lower back pain intensity scores will decrease after taking anti-inflammatory drugs. The table shows the lower back pain intensity scores for 12 patients before and after taking anti-inflammatory drugs for 8 weeks. At , is there enough evidence to support the physician’s claim? (Adapted from Archives of Internal Medicine)

Accepted Answer

All right, hello, everyone. So this question says, a group of researchers believes that a new workout routine helps reduce blood pressure levels. In a study with 18 participants, 13 had lower blood pressure after the program, 4 had higher blood pressure, and 1 showed no change. Using a left-tailed sign test at alpha equals 0.05, what is your decision and what does it mean in context? Here we have 4 different answer choices labeled A through D. So, because this is a sign test, right, we have to remove participants that showed no change from the overall sample size. In this case, only one participant showed no change. So the adjusted sample size N is equal to the original 18, subtracted by 1, which gives you 17. From here we identify the type of test. The reason this is a left tilt test is because the claim is that the blood pressure is going to decrease. Which means that we're testing if the median difference that is the after subtracted by the before, is less than 0. Now to clarify, let's say that the negative signs represent participants with lower blood pressure after the program, and positive signs indicate those with higher blood pressure. So, the number of negative signs in this case is equal to the 13 participants that had lower blood pressure after the program. The number of positives is then equal to 4. From here we compute the test statistic. Now, in a sign test, recall that the test statistic is the smaller of the number of positive or negative signs. In this case, 4 is the smaller of the two numbers, which means that our test statistic is equal to 4. And from here, we have to look for a critical value to compare to the test statistic. So let me actually scroll down to open up some space. Here, we're conducting a left-tailed sign test at alpha equals 0.05, with the adjusted sample size equal to 17. And the number of positive or negative signs follows a binomial distribution. So let's say the number of Signed here X. Follows a binomial distribution. Such that n is equal to 17 and P is equal to 0.5. The reason why P is equal to 0.5 is because according to the null hypothesis, we would expect to see no difference between the before and the after. Which means that Either outcome, whether it's an increase or decrease, is expected to be equally likely. So that's why P is equal to 0.5. And now on to the critical value. In a left-tailed test, we're looking for a small number of negative signs. That would be unusually low if there was really no effect. So, we want to find the largest value of lower case x such that the probability. Of capital X being less than or equal to lowercase x. Is itself less than or equal to alpha, or 0.05. This step of the process is going to involve some trial and error, because the idea is to check for values of X until the cumulative probability crosses 0.05. After this trial and error. You'll notice that the probability of capital X being less than or equal to 4. is approximately 0.0245, which itself is pretty close to 0.05. Just to be thorough, you would also check for the probability that X is less than or equal to 5. But this gives you 0.0717, which is too high because it's greater than 0.05. So The critical value is 4. And specifically, the critical value is less than or equal to 4, which brings us now to the comparison that we must make between the test statistic and the critical value. In this case, recall that our test statistic is equal to 4. Which is equal to our critical value. And because of this, We can reject. The null hypothesis. Now that we know this though, what does it mean for the result of this particular study? Well, rejecting no hypothesis would state that there is sufficient evidence at the 0.05 significance level to support the claim that the new workout routine does in fact reduce blood pressure. So now going back to the answer choices. This corresponds to option B. Which reads as follows. Reject H knot. There is sufficient evidence to support the claim that the workout routine reduces blood pressure. And there you have. So with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Null Hypothesis

Sign Test

P-value

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