A professional golfer wanted to determine if there was a difference in the number of yards that golfers could hit a ball depending on the brand of driver (a particular type of club) used. Twenty-five golfers (of approximately the same ability) were divided into five groups ,and each golfer in a group hit a ball with one particular brand of driver. The number of yards they hit the ball was recorded, and a rank was assigned to each value. The data are tabulated below with the number of yards hit listed first and the ranks listed in parentheses.
$\begin{array}{ccccc}
\hline \text { Brand A } & \text { Brand B } & \text { Brand C } & \text { Brand D } & \text { Brand E } \
\hline 300(25) & 190(5) & 235(11) & 275(17) & 165(2) \
298(23) & 164(1) & 296(21) & 299(24) & 175(3) \
297(22) & 239(13) & 274(16) & 185(4) & 256(14) \
260(15) & 215(7) & 280(18) & 238(12) & 224(9) \
295(20) & 225(10) & 290(19) & 196(6) & 219(8) \
\hline
\end{array}$
Compute the appropriate test value for a Kruskal-Wallis test, at $\alpha = 0.05$, of the professio golfer's hypothesis. onal
A) 13.610
B) 16.610
C) 3.805
D) 6.805

Question

Quizplus · Accepted Answer

The Kruskal-Wallis test statistic is calculated as:$$H = \frac{12}{n(n+1)} \sum_{i=1}^{k} n_i (\bar{r}_i - \bar{r})^2$$where n is the total sample size, k is the number of groups, ni is the sample size of the ith group, $\bar{r}_i$ is the average rank of the ith group, and $\bar{r}$ is the overall average rank.In this case, n = 25, k = 5, n1 = 5, n2 = 5, n3 = 5, n4 = 5, n5 = 5, $\bar{r}_1$ = 15, $\bar{r}_2$ = 6, $\bar{r}_3$ = 14.8, $\bar{r}_4$ = 13.6, and $\bar{r}_5$ = 10.4.Plugging these values into the formula, we get:$$H = \frac{12}{25(25+1)} \sum_{i=1}^{5} 5 (\bar{r}_i - 13)^2$$$$H = \frac{12}{650} (5(15-13)^2 + 5(6-13)^2 + 5(14.8-13)^2 + 5(13.6-13)^2 + 5(10.4-13)^2)$$$$H = \frac{12}{650} (5(4) + 5(-7)^2 + 5(1.8)^2 + 5(0.6)^2 + 5(-2.6)^2)$$$$H = \frac{12}{650} (20 + 49 + 16.2 + 1.8 + 17.56)$$$$H = \frac{12}{650} (104.56)$$$$H = 1.993$$The critical value for the Kruskal-Wallis test with $\alpha = 0.05$ and k = 5 is 11.07. Since our test statistic (1.993) is less than the critical value, we fail to reject the null hypothesis and conclude that there is not a significant difference in the number of yards that golfers can hit a ball depending on the brand of driver used.

A Professional Golfer Wanted to Determine If There Was a Difference