Using the Wilcoxon Signed-Ranks Test
In Exercises 5–8, r...

Question

Using the Wilcoxon Signed-Ranks Test

In Exercises 5–8, refer to the sample data for the given exercises in Section 13-2. Use the Wilcoxon signed-ranks test to test the claim that the matched pairs have differences that come from a population with a median equal to zero. Use a 0.05 significance level.

Exercise 6 “Do Men and Women Talk the Same Amount?”

Accepted Answer

All right, hello, everyone. So this question says a nutritionist wants to determine if a new diet plan affects cholesterol levels. She measures the cholesterol levels in mil milligrams per deciliter of 8 patients before and after following the diet for 3 months. The data are as follows. Using the Wilcoxen sign ranks test at a 0.05 significance level, tests the claim that the matched pairs have differences that come from a population with a median equal to zero. Assume that the population of differences has a distribution that is approximately symmetric. So Recall that for this question, what we're doing is testing the null hypothesis. That is, we're testing if matched pairs are from a population with differences having a median of zero. So, for the Wilcox and sign ranks test, recall that we have to assume that the data comes from a simple random sample, or SRS as I'm writing here for short. In addition to assuming the data is in a simple random sample, We have to Assume, right, that the population of differences has a distribution that is approximately symmetric now here. We're going to assume this as stated in the text of the question, but just keep in mind that this has to be true. The population has to be approximately symmetric for the test to hold. So first we have to begin by finding the difference for each pair. So that's before, subtracted from after. The values for the differences are as follows. We have -8 16 -9. 7 -11, 34. Or 4 rather and -12. So from here We're going to begin by. We're going to begin by taking the absolute value of all of the differences, and then ranking those absolute values from lowest to highest, with one being the lowest. And so after that's done, we're going to replace the differences themselves with the corresponding rank value. And so the ranking Of each absolute value of the difference listed in order of the pairs themselves are as follows we have 4. 8 53. 612, and 7. So again, what was done here is that you first find the absolute value of each difference. Rank them in order from lowest to highest, and then replace the absolute value of the difference itself by the ranking. And now that we have the ranking, we would then attach the original sign of the difference to each ranking. So that's -4. Positive 8 5 Positive 3 6 Positive 1 Positive 2 and -7. Mhm From there You want to calculate the sum of all positive ranks. So that would be 8 added to 31 and 2, which gives you 14. In addition, you also want to add the sum of the absolute values of all negative ranks. So that's the absolute value of -4. -5, 6, and -7. And that gives you 22. So, once again, the two rank sums are 14 and 22. So, here, you take the smaller of the two and allow it to be T. In this case, capital T is going to be equal to 14. So now that we have capital T, recall that lowercase n is the number of pairs of data for which the difference d is not zero. And in this case is going to be equal to 8, because there is no difference of 0 in the dataset given. So now Here Recall that the critical value can be found using the following table, or using the table given as a reference, or sample sizes or for the number of pairs, less than or equal to 30. When referencing this table, you'll notice that the critical value. At alpha equals 0.05, with a two-tailed test and N equals to 8. That critical value is equal to 4. So now, notice, right? RT value, that's 14, is greater than the critical value of 4. And so because Capital T is greater than the critical value, we fail to reject the null hypothesis. Which now brings us to our final answer. And our final answer reads as follows. It says there is insufficient evidence to warrant rejection of the claim that the matched pairs have differences that come from a population with a median equal to zero. 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

Wilcoxon Signed-Ranks Test

Significance Level

Median

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