A consumer advice web site tested a fuel additive. The distance that 12 cars could travel on five gallons of gasoline was recorded without and then with the additive. The results were as follows. $$\begin{array} { l | c c c c c c c c c c c c }
\text { Car } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \
\hline \text { No Additive } & 105 & 139 & 185 & 108 & 187 & 134 & 170 & 161 & 145 & 115 & 190 & 121 \
\text { With additive } & 115 & 157 & 191 & 108 & 197 & 139 & 183 & 168 & 159 & 115 & 193 & 127
\end{array}$$ a) Compute the test statistic
b) Can you conclude that the median distance traveled with the additive difference from the median
Distance traveled with the additive? Use the α = 0.05 level of significance.
A) a) S=9
b) We can conclude that the median distances are different.
B) a) S=12
b) We cannot conclude that the median distances are different.
C) a) S=12
b) We can conclude that the median distances are different.
D) a) S=9
b) We cannot conclude that the median distances are different.

Question

A consumer advice web site tested a fuel additive. The distance that 12 cars could travel on five gallons of gasoline was recorded without and then with the additive. The results were as follows. $$\begin{array} { l | c c c c c c c c c c c c } 
\text { Car } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \
\hline \text { No Additive } & 105 & 139 & 185 & 108 & 187 & 134 & 170 & 161 & 145 & 115 & 190 & 121 \
\text { With additive } & 115 & 157 & 191 & 108 & 197 & 139 & 183 & 168 & 159 & 115 & 193 & 127
\end{array}$$ a) Compute the test statistic
b) Can you conclude that the median distance traveled with the additive difference from the median
Distance traveled with the additive? Use the α = 0.05 level of significance. 
A) a) S=9
b) We can conclude that the median distances are different.
B) a) S=12
b) We cannot conclude that the median distances are different.
C) a) S=12
b) We can conclude that the median distances are different.
D) a) S=9
b) We cannot conclude that the median distances are different.

Quizplus · Accepted Answer

We will use the Wilcoxon signed-rank test to determine if there is a significant difference between the distances traveled with and without the additive.

a) To compute the test statistic, we first find the differences between the distances with and without the additive:

$$\begin{array} { l | c c c c c c c c c c c c } 
\text { Car } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \
\hline \text { Difference } & 10 & 18 & 6 & 0 & 10 & 5 & 13 & 7 & 14 & 0 & 3 & 6 \
\end{array}$$

We then take the absolute value of these differences and rank them from smallest to largest:

$$\begin{array} { l | c c c c c c c c c c c c } 
\text { Difference } & 0 & 0 & 3 & 5 & 6 & 6 & 7 & 10 & 10 & 13 & 14 & 18 \
\hline \text { Rank } & 1.5 & 1.5 & 4 & 5 & 6.5 & 6.5 & 8 & 9.5 & 9.5 & 11 & 12 & 13 \
\end{array}$$

We then find the sum of the ranks for the positive differences (P) and the sum of the ranks for the negative differences (N), and use the smaller sum as the test statistic:

$$P = 9.5 + 11 + 12.5 + 13 = 46$$
$$N = 1.5 + 1.5 + 4 + 5 + 6.5 + 6.5 + 8 + 9.5 + 9.5 = 52$$
$$S = \min(P,N) = 46$$

b) To determine if we can conclude that the median distances are different, we compare S to the critical value from the Wilcoxon signed-rank table for n=12 at the α=0.05 significance level. We find that the critical value is 20, which is much smaller than our test statistic of 46. Therefore, we reject the null hypothesis and conclude that the median distances traveled with and without the additive are different.

A Consumer Advice Web Site Tested a Fuel Additive A) Compute the Test Statistic B) Can You Conclude That

A Consumer Advice Web Site Tested a Fuel Additive A) Compute the Test Statistic
B) Can You Conclude That