To solve for the missing values in the given one-way ANOVA table, we need to apply the standard formulas used in ANOVA calculations.

The ANOVA Table Layout:

Source of Variation	SS	df	MS	F
Between (Treatment)	86.97
Within (Error)	1113.216
Total

Step-by-Step Calculation

1. Degrees of Freedom (df):

   • For the Between (Treatment) df: $df_{\text{Between}} = k - 1$ where $k$ is the number of treatments (groups). Here, $k = 3$. $df_{\text{Between}} = 3 - 1 = 2$

   • For the Within (Error) df: $df_{\text{Within}} = N - k$ where $N$ is the total number of observations across all treatments. Given the sample sizes $n_1 = 15$, $n_2 = 9$, and $n_3 = 11$, we have: $N = 15 + 9 + 11 = 35$ So, $df_{\text{Within}} = 35 - 3 = 32$

   • For the Total df: $df_{\text{Total}} = N - 1 = 35 - 1 = 34$

2. Mean Squares (MS):

   • Mean Square for Between (MS Between) is calculated by: $MS_{\text{Between}} = \frac{SS_{\text{Between}}}{df_{\text{Between}}}$ Given $SS_{\text{Between}} = 86.97$ and $df_{\text{Between}} = 2$: $MS_{\text{Between}} = \frac{86.97}{2} = 43.485$

   • Mean Square for Within (MS Within) is calculated by: $MS_{\text{Within}} = \frac{SS_{\text{Within}}}{df_{\text{Within}}}$ Given $SS_{\text{Within}} = 1113.216$ and $df_{\text{Within}} = 32$: $MS_{\text{Within}} = \frac{1113.216}{32} = 34.81$

3. F-Statistic:

   The F-statistic is calculated by: $F = \frac{MS_{\text{Between}}}{MS_{\text{Within}}}$ Substituting the values for $MS_{\text{Between}} = 43.485$ and $MS_{\text{Within}} = 34.81$: $F = \frac{43.485}{34.81} \approx 1.25$

Complete ANOVA Table:

Source of Variation	SS	df	MS	F
Between (Treatment)	86.97	2	43.485	1.25
Within (Error)	1113.216	32	34.81
Total	1200.186	34

Summary of Calculations:

• df Between (Treatment) = 2
• df Within (Error) = 32
• df Total = 34
• MS Between = 43.485
• MS Within = 34.81
• F-Statistic = 1.25

Would you like more details or clarification? Here are some related questions:

1. How do you interpret the F-statistic in the context of hypothesis testing?
2. What assumptions are made when performing a one-way ANOVA?
3. How do you calculate the p-value from the F-statistic in ANOVA?
4. What does a high F-statistic indicate about the variability between group means?
5. How would you conduct a post-hoc test if the ANOVA result is significant?

Tip: Always check the assumptions of normality and homogeneity of variances before interpreting the results of an ANOVA.