To solve this ANOVA problem step by step, let’s break it down:

Known Data:

• Total Sum of Squares (SS Total): $2,870.0$
• Model Sum of Squares (SS Model): $1,091.9$
• Degrees of Freedom for Groups (df Model): $g - 1 = 8 - 1 = 7$
• Degrees of Freedom for Residuals (df Residual): $g \cdot (n - 1) = 8 \cdot (6 - 1) = 40$
• Total Degrees of Freedom (df Total): $g \cdot n - 1 = 8 \cdot 6 - 1 = 47$

From this information:

• SS Residual = $SS Total - SS Model = 2,870.0 - 1,091.9 = 1,778.1$

Fill the ANOVA Table:

1. Model Mean Square (MS Model):
   $MS Model = \frac{SS Model}{df Model} = \frac{1,091.9}{7} \approx 155.986$

2. Residual Mean Square (MS Residual):
   $MS Residual = \frac{SS Residual}{df Residual} = \frac{1,778.1}{40} \approx 44.453$

Test Statistic $F$:

$F = \frac{MS Model}{MS Residual} = \frac{155.986}{44.453} \approx 3.51$

Effect Size ($\eta^2$):

$\eta^2 = \frac{SS Model}{SS Total} = \frac{1,091.9}{2,870.0} \approx 0.3803$

Pooled Standard Deviation ($\sigma_e$):

The pooled standard deviation is the square root of $MS Residual$:
$\sigma_e = \sqrt{MS Residual} = \sqrt{44.453} \approx 6.67$

P-value:

To determine the P-value, the $F$-distribution must be consulted with $df_1 = 7$ and $df_2 = 40$. Based on these values and $F \approx 3.51$, we can find the approximate $p$-value using an $F$-table or software. I can confirm this more precisely if you prefer!

Let me know if you need any clarifications or more details! 😊

Questions for Further Practice:

1. How would the results change if the number of groups $g$ increased while $n$ remained constant?
2. What assumptions are necessary for performing an ANOVA test?
3. How do you interpret a small $\eta^2$ effect size?
4. What are post-hoc tests, and when should they be applied after an ANOVA?
5. Can you calculate the confidence intervals for group means using the $MS Residual$?

Tip:

Always double-check degrees of freedom calculations to ensure the table is consistent!