Researchers examined the effect of phosphate supplementation on bone formation in six healthy adult dogs. For each dog, bone formation was measured twice: once after 12 weeks of phosphate supplementation and once after a 12-week control period. The results in percent bone growth per year are shown below. Do the data provide evidence that phosphate supplementation significantly stimulates bone formation? $\begin{array} { | l | l | l | l | l | l | l | } \hline \text { Dog ID } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Control } & 1.73 & 3.37 & 3.59 & 2.05 & 1.86 & 3.60 \\\hline \text { Phosphate } & 8.16 & 4.58 & 3.98 & 5.24 & 3.04 & 7.03 \\\hline\end{array}$
The researchers are interested in testing the following one-sided test:
H₀: Bone formation with and without phosphate supplementation have the same distribution.
H_a: Bone formation is systematically higher with phosphate supplementation.
Using the Normal approximation, what is the P-value for the Wilcoxon signed rank test W⁺?

Question

Researchers examined the effect of phosphate supplementation on bone formation in six healthy adult dogs. For each dog, bone formation was measured twice: once after 12 weeks of phosphate supplementation and once after a 12-week control period. The results in percent bone growth per year are shown below. Do the data provide evidence that phosphate supplementation significantly stimulates bone formation?

\begin{array} { | l | l | l | l | l | l | l | } \hline \text { Dog ID } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Control } & 1.73 & 3.37 & 3.59 & 2.05 & 1.86 & 3.60 \\\hline \text { Phosphate } & 8.16 & 4.58 & 3.98 & 5.24 & 3.04 & 7.03 \\\hline\end{array}

The researchers are interested in testing the following one-sided test:
H₀: Bone formation with and without phosphate supplementation have the same distribution.
H_a: Bone formation is systematically higher with phosphate supplementation.
Using the Normal approximation, what is the P-value for the Wilcoxon signed rank test W⁺?

Quizplus · Accepted Answer

We first calculate the signed differences between percent bone growth under phosphate and percent bone growth under control for each dog: $$\begin{array} { | l | l | l | l | l | l | l | } \hline \text { Dog ID } & 1 & 2 & 3 & 4 & 5 & 6 \ \hline \text { Signed difference } & 6.43 & 1.21 & 0.39 & 3.19 & 1.18 & 3.43 \ \hline \end{array}$$ The absolute value of the most negative signed difference is 0.39, which is the absolute value of the minimum of the signed differences. There are 6 dogs, so the approximate P-value for the Wilcoxon signed rank test is $1 - (\frac{{6 \choose 1} + {6 \choose 2}}{2^{6-1}})\frac{1}{2}=0.032$, using the Normal approximation. Since this is a one-sided test, we only reject H₀ in favor of H_a if the P-value is less than 0.05. Therefore, we cannot reject H₀, but we can reject H_a with a 0.032 significance level. The best choice is B: The P-value for the test is between 0.01 and 0.05, so we reject H_a but cannot reject H₀ with a 0.05 significance level.

Researchers Examined the Effect of Phosphate Supplementation on Bone Formation