To fill in the missing within-groups mean square (MS) value in the ANOVA summary table, we need to understand the relationship between the sums of squares (SS), degrees of freedom (df), and mean squares (MS).

ANOVA Summary Table Structure

The ANOVA summary table typically looks like this:

Source	SS	df	MS	F
Between	204	3	68	?
Within	400	8	?
Total	604	11

Steps to Calculate Within-Groups Mean Square (MS):

1. Degrees of Freedom (df):

   • For the between-groups df: $df_{\text{between}} = k - 1$, where $k$ is the number of groups. Here, $k = 4$, so $df_{\text{between}} = 4 - 1 = 3$.
   • For the within-groups df: $df_{\text{within}} = N - k$, where $N$ is the total number of observations. Here, $N = 12$, so $df_{\text{within}} = 12 - 4 = 8$.
   • Total df: $df_{\text{total}} = N - 1 = 12 - 1 = 11$.

2. Mean Square (MS):

   • Mean Square is calculated by dividing the Sum of Squares (SS) by the degrees of freedom (df).
   • For within-groups: $MS_{\text{within}} = \frac{SS_{\text{within}}}{df_{\text{within}}}$.

Given Values:

• $SS_{\text{within}} = 400$
• $df_{\text{within}} = 8$

Calculation:

$MS_{\text{within}} = \frac{SS_{\text{within}}}{df_{\text{within}}} = \frac{400}{8} = 50$

So, the missing within-groups mean square value (MS) is 50.

Updated ANOVA Summary Table:

Source	SS	df	MS	F
Between	204	3	68	?
Within	400	8	50
Total	604	11

Would you like any further details or have any additional questions?

Here are five numbered questions you might want to ask:

1. How do you calculate the F-ratio in an ANOVA table?
2. What are the assumptions underlying ANOVA?
3. How do you interpret a significant F-ratio?
4. What are some common post-hoc tests used after ANOVA?
5. How can you check for homogeneity of variances in ANOVA?

Tip: When conducting an ANOVA, always check the assumptions of normality and homogeneity of variances to ensure the validity of your results.