Question 13.4: Times to Complete an Obstacle Course Two independent random ......
Times to Complete an Obstacle Course
Two independent random samples of army and marine recruits are selected, and the time in minutes it takes each recruit to complete an obstacle course is recorded, as shown in the table. At α = 0.05, is there a difference in the times it takes the recruits to complete the course?
\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Army }& 15 & 18 & 16 & 17 & 13 & 22 & 24 & 17 & 19 & 21 & 26 & 28 \\ \hline \text{Marines }& 14 & 9 & 16 & 19 & 10 & 12 & 11 & 8 & 15 & 18 & 25 & \\ \hline \end{array}
Step 1 State the hypotheses and identify the claim.
H_{0} : There is no difference in the times it takes the recruits to complete the obstacle course.
H_{1} : There is a difference in the times it takes the recruits to complete the obstacle course (claim).
Step 2 Find the critical value. Since \alpha=0.05 and this test is a two-tailed test, use the critical values of +1.96 and -1.96 from Table E.
Step 3 Compute the test value.
a. Combine the data from the two samples, arrange the combined data in ascending order, and rank each value. Be sure to indicate the group.
\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Time }& 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 15 & 16 & 16 & 17 \\ \hline \text{Group }& \mathrm{M} & \mathrm{M} & \mathrm{M} & \mathrm{M} & \mathrm{M} & \mathrm{A} & \mathrm{M} & \mathrm{A} & \mathrm{M} & \mathrm{A} & \mathrm{M} & \mathrm{A} \\ \hline \text{Rank }& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8.5 & 8.5 & 10.5 & 10.5 & 12.5 \\ \hline \end{array}
\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Time }& 17 & 18 & 18 & 19 & 19 & 21 & 22 & 24 & 25 & 26 & 28 \\ \hline \text{Group }& \mathrm{A} & \mathrm{M} & \mathrm{A} & \mathrm{A} & \mathrm{M} & \mathrm{A} & \mathrm{A} & \mathrm{A} & \mathrm{M} & \mathrm{A} & \mathrm{A} \\ \hline \text{Rank }& 12.5 & 14.5 & 14.5 & 16.5 & 16.5 & 18 & 19 & 20 & 21 & 22 & 23 \\ \hline \end{array}
b. Sum the ranks of the group with the smaller sample size. (Note: If both groups have the same sample size, either one can be used.) In this case, the sample size for the marines is smaller.
\begin{aligned}R & =1+2+3+4+5+7+8.5+10.5+14.5+16.5+21 \\& =93\end{aligned}
c. Substitute in the formulas to find the test value.
\begin{aligned}\mu_{R} & =\frac{n_{1}\left(n_{1}+n_{2}+1\right)}{2}=\frac{(11)(11+12+1)}{2}=132 \\\sigma_{R} & =\sqrt{\frac{n_{1} n_{2}\left(n_{1}+n_{2}+1\right)}{12}}=\sqrt{\frac{(11)(12)(11+12+1)}{12}} \\& =\sqrt{264}=16.2 \\z & =\frac{R-\mu_{R}}{\sigma_{R}}=\frac{93-132}{16.2}=-2.41\end{aligned}
Step 4 Make the decision. The decision is to reject the null hypothesis, since -2.41<-1.96.
Step 5 Summarize the results. There is enough evidence to support the claim that there is a difference in the times it takes the recruits to complete the course.