Rank Sums Exercise 12 uses Data Set 33 “Disney World Wait Time...

Question

Rank Sums Exercise 12 uses Data Set 33 “Disney World Wait Times” in Appendix B, and the sample size for the 5:00 PM Tower of Terror wait times is n = 50.

b. If we have sample paired data with 50 nonzero differences, what is the expected value of T if the population consists of matched pairs with differences having a median of 0?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A researcher collects pair data with 36 non-zero differences, assuming the population of matched pairs has differences with a median of zero, what is the expected value of T in the Wilcoxson signed ranks test? Awesome. So it appears for this particular prong when we are focusing our attention on this researcher that has collected paired data with 36 non-zero differences, and we're asked to assume the population of matched pairs has differences with a meeting of zero, we're asked to ultimately determine what is the expected value of T. In the Wilcoxson signed ranks test. So now that we know that we're ultimately trying to figure out what this expected value of T is, let us read off our multiple choice answers to see what our final answer might be. A is 167, B is 333, C is 666. And D is 1,332. Awesome. So first off, let us recall and note that in the Wilcoxson signed ranks test, T is the smaller of the sum of positive ranks and the sum of negative ranks. So let us note that we are told from the problem itself we are given that the number of non-zero differences, we'll call this value N. is equal to 36, and the null hypothesis, which will denote this as H 0, and the null hypothesis states that the median of the differences is zero, so positive and negative differences are equally likely. So now we can focus on our attention on trying to determine the expected value of tea. So, under the null hypothesis, each difference is equally likely to be positive or negative, so the sum of positive ranks and the sum of negative ranks are on average equal. So therefore we could go ahead and write that the ranks from 1 to 36 sum 2. 1 + 2 plus continuing onward 2 + 36 will be equal to 36 multiplied by 37, so 36 multiplied by 37 divided by 2, which will be equal to 666. And the sum from 1 to 36 can also be obtained using the arithmic series formula which states that, as we should recall, S subscript N is equal to N multiplied by A1 plus A N all divided by 2. Awesome. And as we should note, A1 denotes the first term, AN denotes the last term, and N denotes the number of terms being added. So when we plug in our known variables, we will find that SN is equal to, or in this case, N, in this case is 36, so we could say that S subscript 36 is equal to 36 multiplied by parentheses 1 plus. 36 divided by 2, which will be also equal to 666. Fantastic. So on average, half are positive and half are negative. So the expected sum of positive ranks is equal to the expected sum of negative ranks, which is going to be equal to 666 divided by 2, which is going to be equal to 333. And this is our final answer. Hooray, we did it. So as we should recall a note, T is defined as the smaller of these two sums, but the expected sums of positive and negative ranks are equal. So under the null hypothesis, the expected value of T. Without a doubt, will be 333, and this corresponds with multiple choice answer letter B, which once again is 333. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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Key Concepts

Paired Data

Median of Differences

Expected Value of T

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