Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to perform the Wilcoxon signed ranks test for earnings by education. Here's our problem statement: The table shows the earnings and thousands of dollars of a random sample of 11 people with bachelor's degrees and 10 people with associate's degrees, and alpha equals 0.05. Is there enough evidence to support the belief that there is a difference in the earnings of people with bachelor's degrees and those with associate's degrees? Complete Parts A through E below.
OK, first we're asked to write the claim mathematically and identify the null and alternative hypotheses. We know that the null hypothesis is going to be a statement of equality. And here it's pretty much going to be that there is no difference in the earnings. The alternative hypothesis will then be that there is a difference in the earnings. Let's check what we're testing here. I don't see anything about a claim but we're looking for evidence to support the belief that there is a difference in the earnings of people with bachelor's degrees and those with associate's degrees. So there is a difference in the earnings, and that's going to be the claim that we're testing. And that's going to be the alternative hypothesis. Nice work!
Now Part B asks us for the critical values. We can get the critical values from the distribution calculator in StatCrunch. Our test statistic is a Z score, and so therefore we can get our critical values from the standard Normal distribution because that's where Z scores come from --- the standard Normal distribution. So here in StatCrunch, I'm going to go to Stat --> Calculators --> Normal. Here my alternative hypothesis is there is a difference. So that difference could be less than or greater than; it could be negative or positive. So that means that we're going to have to select the Between option up here because we've got a two-tailed test. And let's see, we've got a significance level of 5%, so that means the area in between the tails is going to be 95%. And there's my critical values. I'm gonna use this plus or minus sign so that I don't have to type the number more than once. Excellent!
Now Part C asks us for the test statistic. The test statistic --- well, I would love it if StatCrunch could do this, but for an independent Wilcoxon signed ranks test, that's not going to happen, at least not with the way StatCrunch is coded right now. So that means we're going to have to go the old school route and use the data in Excel.
Here's my data in Excel. Let's do a little bit of housekeeping so we can see a little bit better everything that's going on. First thing I'm going to do is I'm going to center everything. And now we're going to come and make that whole first row bold typeface. Let me go ahead and replace some of these values here. We'll have a sample row here for our first column. So here we're going to list the numbers. I'm going to solve with the bachelors salary is going to be Sample 1, and then my associates salary is going to be Sample 2. No, we don't need you anymore. Now we're just going to relabel you as salary. OK, now we can see better everything that's going on.
So now in the next column I want to put my rankings, but I've got to sort this list first. So let's go up to Data --> Sort. I want to sort by salary from smallest to largest. Now we can put our rankings in the preliminary rankings. We'll start with just one, two, three, and so on and so forth. There we go. Now we've got 21 values in our data set.
And so now we want to look to see if there are any same salaries. And there are, so we've got to break ties, which means we've got to make adjustments to our rankings. So let's see, the first tie is right here, so that average is 4.5. So that changes the rankings of those to 4.5. And then these two are the same, so we have to replace those. And let's see, what else did we get? Those three are the same, so we're going to replace those. And the next two are the same, so we replaced those. And then these two are the same, so we're going to replace those. And that's it. So now the ones that have unique values to them, we're just going to have to bring those over, and there. We're done. So now we've got all of the proper rankings that we have for each of our values.
Now I'm going to resort my list, this time by sample number. So now I've got all the samples here, and our value is going to be some of the smaller samples. So we see Sample 1. It has a sum of 143, but there's 11 here. The other sample has 10 values, and its sum is 88. So this is going to be our R-value 88. n1 will be 10, and then n2 will be 11 . So to start off with our old school calculation here, here's our formula. And we've got 88 for an R-value, n1 is 10 and n2 is 11. So we substitute these values into our expression, and then we start to simplify. So we get to a point where we can put stuff in a calculator and out comes our test statistic, which we're asked to round to two decimal places. Excellent!
Part D says, "Decide whether to reject or fail to reject the null hypothesis." We can do that. Simple enough. We've got our test statistic here. We've got our critical values here . So if I come back here to StatCrunch, we've got the critical value here in the tails and they're bounded by --- excuse me, the critical values are bounding the tails. And we've got our test statistic, -1.55, so that's going to put us right about here. So our test statistic is here in between the tails outside the region of rejection. So because it's between the critical values and outside the reason of rejection, we're going to fail to reject the null hypothesis. Good job!
Now Part E wants us to interpret this conclusion of our hypothesis test in the context of the original claim. So we failed to reject the null hypothesis. That means it could potentially be true, and failing to reject the null hypothesis means that this statement "There is no difference in earnings" is potentially true. So there's not enough evidence to support the claim. Well done!
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