We are comparing the effects of a new medication, an existing medication, and a placebo on patient outcomes. We are given the sample means and sample sizes for each group, as well as the between-groups sum of squares ($SS_B$) and the within-groups sum of squares ($SS_W$). We need to calculate the F-statistic to test for differences between the groups.

1. Problem Description

We are comparing the effects of a new medication, an existing medication, and a placebo on patient outcomes. We are given the sample means and sample sizes for each group, as well as the between-groups sum of squares ($SS_B$) and the within-groups sum of squares ($SS_W$). We need to calculate the F-statistic to test for differences between the groups.

2. Solution Steps

First, we need to calculate the degrees of freedom for the between-groups variance ($df_B$) and the within-groups variance ($df_W$).

$df_B = k - 1$, where $k$ is the number of groups. In this case, $k = 3$.

$df_W = N - k$, where $N$ is the total sample size. In this case, $N = n_1 + n_2 + n_3 = 20 + 43 + 21 = 84$.

$df_B = 3 - 1 = 2$

$df_W = 84 - 3 = 81$

Next, we need to calculate the mean square between groups ($MS_B$) and the mean square within groups ($MS_W$).

$MS_B = \frac{SS_B}{df_B} = \frac{47.31}{2} = 23.655$

$MS_W = \frac{SS_W}{df_W} = \frac{2024.71}{81} = 25$ (approximately)

Finally, we can calculate the F-statistic:

$F = \frac{MS_B}{MS_W} = \frac{23.655}{25} = 0.9462$

3. Final Answer

The F-statistic is approximately 0.

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