Question

In the introduction to this chapter, we considered a study comparing three groups of college students (soccer athletes, nonsoccer athletes, and a control group consisting of students who did not participate in intercollegiate sports). The following information on scores from the Hopkins Verbal Learning Test (which measures immediate memory recall) was $$\begin{array}{l|ccc} \text { Group } & \text { Soccer Athletes } & \text { Nonsoccer Athletes } & \text { Control } \\ \hline \text { Sample size } & 86 & 95 & 53 \\ \text { Sample mean score } & 29.90 & 30.94 & 29.32 \\ \begin{array}{l} \text { Sample standard } \\ \text { deviation } \end{array} & 3.73 & 5.14 & 3.78 \\ \hline \end{array}$$ In addition, \(\overline{\bar{x}}=30.19\). Suppose that it is reasonable to regard these three samples as random samples from the three student populations of interest. Is there sufficient evidence to conclude that the mean Hopkins score is not the same for the three student populations? Use \(\alpha=.05\).

Short answer

Based on the comparison of the calculated F statistic to the critical F value, it can be determined whether there is enough evidence to reject the null hypothesis that all population means are equal and conclude that there is a statistically significant difference among the mean Hopkins scores of the Soccer athletes, Nonsoccer athletes, and control group.

Step-by-step solution

Calculate the Group Totals

For each group, multiply the sample mean by the sample size. For the Soccer Athletes group, this is \(29.90 * 86 = 2571.4\). For the Nonsoccer Athletes group, this is \(30.94 * 95 = 2939.3\). And for the control group, it is \(29.32 * 53 = 1552.96\)

Calculate the overall Total

You then sum up all the group totals to get the overall total. This is \(2571.4 + 2939.3 + 1552.96 = 7063.66\)

Calculate Sum of Squares Total (SSTO)

SSTO is the sum of the squares of the deviations between each individual and the general mean. Using the general mean \(\overline{\bar{x}}= 30.19\), the SSTO can be calculated.

Calculate Sum of Squares Between (SSB)

SSB is the sum of squares of the deviations between each group mean and the general mean, each multiplied by the size of that group. Use the means and sample sizes of each group to calculate SSB.

Calculate Sum of Squares Within (SSW)

SSW is the sum of the squares of the deviations within each group, which is simply SSTO - SSB.

Calculate degrees of freedom

The degrees of freedom for SSB (dfB) is the number of groups minus 1 (\(g - 1\)), for this example 2. The degrees of freedom for SSW (dfW) is the total number of observations minus the number of groups (\(n - g\))

Calculate Mean Square Between (MSB) and Mean Square Within (MSW)

MSB is SSB divided by its degrees of freedom and MSW is SSW divided by its degrees of freedom.

Compute F Statistic

F statistic is the ratio of MSB to MSW.

Compare to F Critical Value

Once we have found the F statistic, we can compare it with the critical F value for a given alpha level (0.05) and degrees of freedom. If the computed F is larger than the critical F, then we reject the null hypothesis that all population means are equal.