A State Environmental Board wants to compare pollution levelsin two of its major

Question

A State Environmental Board wants to compare pollution levelsin two of its major cities. Sunshine City thrives on the tourist industry and Service City thrives on the service industry. The Environmental Board randomly selects several areas within the cities and measures the pollution levels in the parts per million with the following results:

a. Using the Wilcoxon Rank-Sum Test, does the data substantaite the claim that the Brand A amplifier has a higher median maximim power output than Brand B at a=0.05?

b. What assumptions were made in performing the test in part a?

3. A State Environmental Board wants to compare pollution levels in two of its major cities. Sunshine City tourist industry and Service City thrives on the service industry. The Environmental Board randomly selects several areas within the cities and measures the pollution levels in parts per million with the following results: a. Using the Wilcoxon Rank-Sum Test, does the data substantiate the claim that the Brand A amplifier has a higher median maximum power output than Brand B at 0.05? b. What assumptions were made in performing the test in part a? Pollution Levels (in ppm) Service City Rank Rank Sunshine City 780 800 805 828 755 772 807 830 753 770 803 826 757 774

Explanation / Answer

Set up hypotheses as

H0: The median difference is zero versus

H1: The median difference is greter than zero

The test statistic for the Wilcoxon Signed Rank Test is W, defined as the smaller of W+ and W- which are the sums of the positive and negative ranks, respectively.

so we first find the difference between them and finding rank of difference so we have

calculating W+ and W- but here all d values are positive so W-=0 and W+=sum of ranks =2+2+2+4+6+6+6 =28

Recall that the sum of the ranks (ignoring the signs) will always equal n(n+1)/2. i.e n(n+1)/2 =7*8/2 =28

W =max(W+,W-) =(28,0) =28

find critical values of W in wilcoxon signed rank test table , which is W(critical) =2 so 28>2 and p value is 0.021(<0.05) so we reject H0 at 0.05 level of significance so the Brand A amplifier has a higher median maximim power output than Brand B.

A B d=A-B ordered d Rank d sign 800 780 20 17 2 + 828 805 23 17 2 + 772 755 17 17 2 + 830 807 23 20 4 + 770 753 17 23 6 + 826 803 23 23 6 + 774 757 17 23 6 +

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