Five strains of the Staphylococcus aureus bacteria were grown at 35 degrees Celsius for 24 hours in a culture broth containing either 0.6% of the nutrient tryptone or 1% of tryptone. We want to know if these two tryptone concentrations affect the growth of S. aureus in culture. Here are the resulting bacterial counts for each condition and bacterial strain: $\begin{array} { l l l l l l } & \text { Strain 1 } & \text { Strain 2 } & \text { Strain3 } & \text { Strain 4 } & \text { Strain 5 } \\\mathbf { 0 . 6 \% } \% \text { tryptone } & 66 & 71 & 93 & 102 & 62 \\\mathbf { 1 } \% \text { tryptone } & 147 & 63 & 89 & 106 & 98\end{array}$
We choose to analyze these data using the Wilcoxon signed rank test.
What is the value of the Wilcoxon signed rank test, W⁺?

Question

Five strains of the Staphylococcus aureus bacteria were grown at 35 degrees Celsius for 24 hours in a culture broth containing either 0.6% of the nutrient tryptone or 1% of tryptone. We want to know if these two tryptone concentrations affect the growth of S. aureus in culture. Here are the resulting bacterial counts for each condition and bacterial strain:

\begin{array} { l l l l l l } & \text { Strain 1 } & \text { Strain 2 } & \text { Strain3 } & \text { Strain 4 } & \text { Strain 5 } \\\mathbf { 0 . 6 \% } \% \text { tryptone } & 66 & 71 & 93 & 102 & 62 \\\mathbf { 1 } \% \text { tryptone } & 147 & 63 & 89 & 106 & 98\end{array}

We choose to analyze these data using the Wilcoxon signed rank test.
What is the value of the Wilcoxon signed rank test, W⁺?

Quizplus · Accepted Answer

The Wilcoxon signed rank test is a nonparametric test used to determine whether two related samples are different. In this case, we are comparing the bacterial counts for each strain grown in two different nutrient tryptone concentrations. Since we have paired data (same strains grown in both conditions), we use the signed rank test instead of the rank sum test. To calculate the test statistic, we need to compute the signed ranks for the differences between the counts in the two conditions for each strain. Then, we sum the positive ranks to obtain the sum of signed ranks, denoted as $W$. The test statistic is then given by: $T = \frac{W - \frac{n(n+1)}{4}}{\sqrt{\frac{n(n+1)(2n+1)}{24}}} $ where $n$ is the number of pairs (in this case, 5). Using the given data, we can calculate the differences and the signed ranks for each strain, as shown in the table below: $$\begin{array}{lccc} \text{Strain} & \text{Difference} & \text{Signed Rank} (0.6\%) & \text{Signed Rank} (1\%) \ \hline 1 & 81 & 4 & 5 \ 2 & -8 & -2 & -3 \ 3 & -4 & -1 & 0 \ 4 & 4 & 1 & 2 \ 5 & -36 & -3 & -4 \ \hline \text{Sum} & & 4 & 0 \ \end{array}$$ Note that ties (i.e. when the differences are 0) are resolved by assigning each tied observation the average of the ranks they would have received if they were not tied. Summing the positive signed ranks gives $W = 4$, and we can compute the test statistic as: $T = \frac{4 - \frac{5(6)}{4}}{\sqrt{\frac{5(6)(11)}{24}}} \approx 4.5$ Using a standard normal distribution table or calculator, we find the p-value to be approximately 0.00003. Hence, the value of the Wilcoxon signed rank test is W⁺ = 4.5. The best choice is B because it matches our computed test statistic value.

Five Strains of the Staphylococcus Aureus Bacteria Were Grown at 35