# Given data from the problem
mean_simple = 30.64
std_dev_simple = 20.04
n_simple = 50

mean_elaborate = 26.21
std_dev_elaborate = 23.70
n_elaborate = 42

mean_pairwise = 15.13
std_dev_pairwise = 15.70
n_pairwise = 47

# Total number of samples
n_total = n_simple + n_elaborate + n_pairwise

# Overall mean (weighted mean of all groups)
overall_mean = (mean_simple * n_simple + mean_elaborate * n_elaborate + mean_pairwise * n_pairwise) / n_total

# Calculating Sum of Squares for Treatments (Between-group variability)
ss_treatments = n_simple * (mean_simple - overall_mean)**2 + \
                n_elaborate * (mean_elaborate - overall_mean)**2 + \
                n_pairwise * (mean_pairwise - overall_mean)**2

# Calculating Sum of Squares for Error (Within-group variability)
ss_error_simple = (std_dev_simple**2) * (n_simple - 1)
ss_error_elaborate = (std_dev_elaborate**2) * (n_elaborate - 1)
ss_error_pairwise = (std_dev_pairwise**2) * (n_pairwise - 1)

ss_error = ss_error_simple + ss_error_elaborate + ss_error_pairwise

# Total Sum of Squares
ss_total = ss_treatments + ss_error

# Degrees of freedom
df_treatments = 2  # 3 groups - 1
df_error = n_total - 3  # Total N - number of groups
df_total = df_treatments + df_error

# Mean Squares
ms_treatments = ss_treatments / df_treatments
ms_error = ss_error / df_error

# F-statistic
f_statistic = ms_treatments / ms_error

# Displaying the calculated results
ss_treatments, ss_error, ss_total, ms_treatments, ms_error, f_statistic, df_treatments, df_errorHere are the full ANOVA calculations based on the provided dataHere are the full ANOVA calculations based on the provided 

- **Sum of Squares for Treatments (SS Treatments)**: 6106.97
- **Sum of Squares for Error (SS Error)**: 54046.31
- **Total Sum of Squares (SS Total)**: 60153.28
- **Mean Square for Treatments (MS Treatments)**: 3053.49
- **Mean Square for Error (MS Error)**: 397.40
- **F-statistic**: 7.68

### Degrees of Freedom:
- **Degrees of Freedom for Treatments**: 2
- **Degrees of Freedom for Error**: 136

### Conclusion:
The F-statistic is 7.68. To determine whether this value is statistically significant, compare it to the critical F-value from an F-distribution table at \( \alpha = 0.05 \), with 2 and 136 degrees of freedom. If the calculated F-value exceeds the critical value, we reject the null hypothesis, indicating that the mean percentages of names recalled differ significantly between the groups.

Let me know if you'd like to continue with any specific post-hoc analysis, or if you need more details!

Here are 5 additional questions:
1. How do you interpret the F-statistic in an ANOVA?
2. What are the implications if the null hypothesis is rejected in this ANOVA?
3. How can we check if the assumptions of ANOVA (normality and homogeneity of variance) hold?
4. What follow-up tests can be conducted after rejecting the null hypothesis in ANOVA?
5. How does increasing the sample size affect the power of the ANOVA test?

**Tip:** Always visualize the data using boxplots or histograms before performing ANOVA to ensure no major deviations from normality.

This page provides a detailed solution for calculating ANOVA, including the sum of squares for treatments and error, the overall mean, and the F-statistic based on sample data. The example uses a dataset with three groups: simple, elaborate, and pairwise, analyzing their mean differences using statistical methods. Suitable for students and professionals learning ANOVA.

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