Question

In this exercise involving paired differences, consider that it is reasonable to assume the populations being compared have approximately the same shape and that the distribution of paired differences is approximately symmetric. Two fuel additives are tested to determine their effect on miles per gallon for passenger cars. Suppose test results for 12 cars follow; each car was tested with both fuel additives. Additive Car 1 1 | 2 20.12 19.05 23.56 20.77 2 3 21.03 22.57 4 20.15 17.06 5 20.2321.22 6 24.77 22.80 7 17.16 18.20 8 19.55 15.98 9 20.87 20.03 10 25.23 21.15 11 24.21 | 22.78 12 24.02 22.70 Use a = 0.05 and the Wilcoxon signed-rank test to see whether there is a significant difference between the median miles per gallon for the additives. State the null and alternative hypotheses. o Ho: Median for additive 1 - Median for additive 2 2 0 Ho: Median for additive 1 - Median for additive 2 < 0 o Ho: Median for additive 1 - Median for additive 2 = 0 M: Median for additive 1 - Median for additive 2 + 0 O Ho: Median for additive 1 - Median for additive 2 s 0 H: Median for additive 1 - Median for additive 2 > 0 Ho: Median for additive 1 - Median for additive 2 + 0 Ha: Median for additive 1 - Median for additive 2 = 0 O Ho: Median for additive 1 - Median for additive 2 > 0 Ho: Median for additive 1 - Median for additive 2 = 0 Find the value of the test statistic. it Find the p-value. (Round your answer to four decimal places.) D-value = State your conclusion. Do not reject Ho. There is sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives. o Do not reject Ho. There is not sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives. O Reject Ho. There is sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives. o Reject M. There is not sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives.

Answer 1

o Ho: Median for additive 1 - Median for additive 2 = 0 Ha: Median for additive 1 - Median for additive 2 + 0

score Patient 1 2 Rank signed rank 4 4 9 9 7 -7 3 10 -2 4 5 6 7 8 9 10 11 12 median before(b) after(a) 20.12 19.05 23.56 20.77 21.03 22.57 20.15 17.06 20.23 21.22 24.77 22.8 17.16 18.2 19.55 15.98 20.87 20.03 25.23 21.15 24.21 22.78 24.02 22.7 d=b-a 1.07 2.79 -1.54 3.09 -0.99 1.97 -1.04 3.57 0.84 4.08 1.43 1.32 d 1.07 2.79 1.54 3.09 0.99 1.97 1.04 3.57 0.84 4.08 1.43 1.32 10 2 8 3 11 1 12 6 5 8 -3 11 1 12 6 5 sample size n = 12 sum of positive ranks - W+ = sum of negative ranks =EW-= 66 12

T=max(W+,W-)       =

66

	mean =μ=n(n+1)/4 =		39
std dev=σ=√n(n+1)(2n+1)/24=			12.75
test statistic z score=(W-μ)/σ =			2.08

p value =

0.0376

from excel: 2*normsdist(-abs(2.08)

o Reject Hy. There is sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives.