Answer :
Alright, let's work through this problem step-by-step:
### Step 1: Compute the Degrees of Freedom (df)
- Between Groups: The degrees of freedom for between groups is calculated as the number of conditions minus 1. Since there are 3 conditions:
[tex]\[ \text{df}_{\text{between}} = 3 - 1 = 2 \][/tex]
- Within Groups (Error): The degrees of freedom within each group is calculated as the number of participants per condition minus 1, multiplied by the number of conditions. With 12 participants per condition:
[tex]\[ \text{df}_{\text{within}} = 3 \times (12 - 1) = 3 \times 11 = 33 \][/tex]
- Total: The total degrees of freedom is the sum of the between groups and within groups degrees of freedom:
[tex]\[ \text{df}_{\text{total}} = \text{df}_{\text{between}} + \text{df}_{\text{within}} = 2 + 33 = 35 \][/tex]
### Step 2: Compute the Sum of Squares (SS)
- Between Groups: The sum of squares for between groups is found using the mean squares (MS) and the degrees of freedom. We know that:
[tex]\[ MS_{\text{between}} = 9 \][/tex]
Therefore:
[tex]\[ SS_{\text{between}} = MS_{\text{between}} \times \text{df}_{\text{between}} = 9 \times 2 = 18 \][/tex]
- Within Groups: The sum of squares within groups is the total sum of squares minus the sum of squares between groups. Given that the total SS is 117:
[tex]\[ SS_{\text{within}} = SS_{\text{total}} - SS_{\text{between}} = 117 - 18 = 99 \][/tex]
### Step 3: Compute the Mean Squares (MS)
- Within Groups (Error): The mean squares within groups is the sum of squares within groups divided by its degrees of freedom:
[tex]\[ MS_{\text{within}} = \frac{SS_{\text{within}}}{\text{df}_{\text{within}}} = \frac{99}{33} = 3.0 \][/tex]
### Step 4: Compute the F-Statistic
The F-statistic is the ratio of the mean squares between groups to the mean squares within groups:
[tex]\[ F = \frac{MS_{\text{between}}}{MS_{\text{within}}} = \frac{9}{3.0} = 3.0 \][/tex]
### Step 5: Hypotheses
- Null Hypothesis [tex]\(H_0\)[/tex]: There is no difference between the means of the conditions (μ₁ = μ₂ = μ₃).
- Alternative Hypothesis [tex]\(H_1\)[/tex]: At least one mean is different from the others.
### Step 6: F-critical Value
The F-critical value is derived from the F-distribution tables based on the degrees of freedom (df_between and df_within) and the chosen significance level (typically α = 0.05):
- df_between = 2
- df_within = 33
This value would typically be looked up in the F-distribution tables or using statistical software.
### Step 7: Decision Rule
- Reject the null hypothesis [tex]\(H_0\)[/tex] if the calculated F-statistic is greater than the F-critical value from the table. Otherwise, do not reject [tex]\(H_0\)[/tex].
### Step 8: Sentence Write-up
This part will depend on whether the calculated F-statistic exceeds the F-critical value. Let's state it generically:
"There is (or isn't, depending on the decision) a significant difference between the treatment conditions."
#### Completed Table:
\begin{tabular}{|l|l|l|l|l|}
\hline Source of Variability & SS & df & MS & F \\
\hline Between Groups & 18 & 2 & 9 & 3.0 \\
\hline Within Groups (Error) & 99 & 33 & 3.0 & \\
\hline Total & 117 & 35 & & \\
\hline
\end{tabular}
#### Summary:
a. Hypotheses:
- [tex]\(H_0: \mu_1 = \mu_2 = \mu_3\)[/tex] (No difference between means)
- [tex]\(H_1: \)[/tex] At least one mean is different
b. F-critical: This value is determined based on F-tables for df_between = 2 and df_within = 33 at the chosen significance level (e.g., α = 0.05).
c. Decision about the null hypothesis: Reject [tex]\(H_0\)[/tex] if F-statistic > F-critical, otherwise do not reject [tex]\(H_0\)[/tex].
d. Sentence write-up: There is (or isn't, depending on the decision) a significant difference between the treatment conditions.
### Step 1: Compute the Degrees of Freedom (df)
- Between Groups: The degrees of freedom for between groups is calculated as the number of conditions minus 1. Since there are 3 conditions:
[tex]\[ \text{df}_{\text{between}} = 3 - 1 = 2 \][/tex]
- Within Groups (Error): The degrees of freedom within each group is calculated as the number of participants per condition minus 1, multiplied by the number of conditions. With 12 participants per condition:
[tex]\[ \text{df}_{\text{within}} = 3 \times (12 - 1) = 3 \times 11 = 33 \][/tex]
- Total: The total degrees of freedom is the sum of the between groups and within groups degrees of freedom:
[tex]\[ \text{df}_{\text{total}} = \text{df}_{\text{between}} + \text{df}_{\text{within}} = 2 + 33 = 35 \][/tex]
### Step 2: Compute the Sum of Squares (SS)
- Between Groups: The sum of squares for between groups is found using the mean squares (MS) and the degrees of freedom. We know that:
[tex]\[ MS_{\text{between}} = 9 \][/tex]
Therefore:
[tex]\[ SS_{\text{between}} = MS_{\text{between}} \times \text{df}_{\text{between}} = 9 \times 2 = 18 \][/tex]
- Within Groups: The sum of squares within groups is the total sum of squares minus the sum of squares between groups. Given that the total SS is 117:
[tex]\[ SS_{\text{within}} = SS_{\text{total}} - SS_{\text{between}} = 117 - 18 = 99 \][/tex]
### Step 3: Compute the Mean Squares (MS)
- Within Groups (Error): The mean squares within groups is the sum of squares within groups divided by its degrees of freedom:
[tex]\[ MS_{\text{within}} = \frac{SS_{\text{within}}}{\text{df}_{\text{within}}} = \frac{99}{33} = 3.0 \][/tex]
### Step 4: Compute the F-Statistic
The F-statistic is the ratio of the mean squares between groups to the mean squares within groups:
[tex]\[ F = \frac{MS_{\text{between}}}{MS_{\text{within}}} = \frac{9}{3.0} = 3.0 \][/tex]
### Step 5: Hypotheses
- Null Hypothesis [tex]\(H_0\)[/tex]: There is no difference between the means of the conditions (μ₁ = μ₂ = μ₃).
- Alternative Hypothesis [tex]\(H_1\)[/tex]: At least one mean is different from the others.
### Step 6: F-critical Value
The F-critical value is derived from the F-distribution tables based on the degrees of freedom (df_between and df_within) and the chosen significance level (typically α = 0.05):
- df_between = 2
- df_within = 33
This value would typically be looked up in the F-distribution tables or using statistical software.
### Step 7: Decision Rule
- Reject the null hypothesis [tex]\(H_0\)[/tex] if the calculated F-statistic is greater than the F-critical value from the table. Otherwise, do not reject [tex]\(H_0\)[/tex].
### Step 8: Sentence Write-up
This part will depend on whether the calculated F-statistic exceeds the F-critical value. Let's state it generically:
"There is (or isn't, depending on the decision) a significant difference between the treatment conditions."
#### Completed Table:
\begin{tabular}{|l|l|l|l|l|}
\hline Source of Variability & SS & df & MS & F \\
\hline Between Groups & 18 & 2 & 9 & 3.0 \\
\hline Within Groups (Error) & 99 & 33 & 3.0 & \\
\hline Total & 117 & 35 & & \\
\hline
\end{tabular}
#### Summary:
a. Hypotheses:
- [tex]\(H_0: \mu_1 = \mu_2 = \mu_3\)[/tex] (No difference between means)
- [tex]\(H_1: \)[/tex] At least one mean is different
b. F-critical: This value is determined based on F-tables for df_between = 2 and df_within = 33 at the chosen significance level (e.g., α = 0.05).
c. Decision about the null hypothesis: Reject [tex]\(H_0\)[/tex] if F-statistic > F-critical, otherwise do not reject [tex]\(H_0\)[/tex].
d. Sentence write-up: There is (or isn't, depending on the decision) a significant difference between the treatment conditions.
