To solve the problem step by step, let's start by filling in the incomplete Analysis of Variance (ANOVA) table.
1.1 Complete the ANOVA Table
Given data:
- Total Sum of Squares (SS_total) = 15.8333
- SS_treatments = 12.5138
- SS_errors = A
- df_total = 14
From the formula: \[ \text{SS}{\text{total}} = \text{SS}{\text{treatments}} + \text{SS}{\text{errors}} \] We can calculate \(A\): \[ A = \text{SS}{\text{total}} - \text{SS}_{\text{treatments}} \] \[ A = 15.8333 - 12.5138 = 3.3195 \]
Next, we need the degrees of freedom (df). Assuming \(k\) is the number of treatment groups and \(n\) is the total sample size, we can determine:
- Let \(k\) be the number of treatments.
- Therefore, the degrees of freedom for treatments \(df_t = k - 1\).
- \(df_e = n - k\) for errors.
- Thus, total degrees of freedom is \(df_{\text{total}} = n - 1 = df_t + df_e\).
As we know: \[ df_{\text{total}} = 14 \] \[ df_t + df_e = 14 \]
Since we have ss_treatments, let’s denote: Let \(k - 1 = B\), thus \(B\) corresponds to df for treatments. It implies \(df_e = 14 - B\).
From \(SS\) and \(df\):
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Mean square for Treatments (MS): \[ MS_t = \frac{SS_t}{df_t} = \frac{12.5138}{B} \]
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Mean square for Errors (MS): \[ MS_e = \frac{SS_e}{df_e} = \frac{3.3195}{14 - B} \]
The F-statistic can be calculated as: \[ F = \frac{MS_t}{MS_e} \]
To fill in the table, let’s assume there are 3 treatment levels (i.e., \(k = 3\)): Then:
- \(df_t = 3 - 1 = 2\)
- \(df_e = 14 - 2 = 12\)
Now we can compute:
-
Compute MS_t: \[ MS_t = \frac{12.5138}{2} = 6.2569 \]
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Compute MS_e: \[ MS_e = \frac{3.3195}{12} = 0.2766 \]
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Compute F-statistic: \[ F = \frac{6.2569}{0.2766} \approx 22.61 \]
Completed ANOVA Table
| Source | SS | df | MS | F | |-------------|----------|----|----------|--------| | Treatments | 12.5138 | 2 | 6.2569 | 22.61 | | Errors | 3.3195 | 12 | 0.2766 | | | Total | 15.8333 | 14 | | |
1.2 Test for the Difference Among the Treatment Means at α = 0.01
To test the differences in treatment means, we find the critical value of \(F\) for \(df_t = 2\) and \(df_e = 12\) at a significance level of \( \alpha = 0.01 \).
Using an F-table or statistical software for \(F(2, 12)\):
- Critical value for \(F_{0.01}(2, 12) \approx 6.93\) (critical value may slightly differ based on tables or software).
Since \(F_{calculated} = 22.61 > 6.93\), we reject the null hypothesis.
Conclusion: There are statistically significant differences among the treatment means at α = 0.01.
1.3 Construct a 99% Confidence Interval for the Mean Differences
We can use the formula to calculate the confidence interval for the difference between two means: \[ CI = (Y̅_1 - Y̅_2) \pm t_{\alpha/2} \times SE \] Where:
- \(Y̅_1 = 8.875\) and \(Y̅_2 = 6.7\)
- \(n_1 = 4\) and \(n_2 = 6\)
- To assume equal variances, we can calculate the pooled standard deviation (SD).
First, we compute the variance components: \(S_1^2\) and \(S_2^2\) are not provided but can be calculated maximally if we had sample variances. We will estimate the pooled variance based on given data as we don’t have sample standard deviations for each treatment's observations.
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Calculate the difference \(Y̅_1 - Y̅_2\): \[ Y̅_1 - Y̅_2 = 8.875 - 6.7 = 2.175 \]
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For constructing a CI, we compute the standard error (SE): \[ SE = \sqrt{ \frac{S_p^2}{n_1} + \frac{S_p^2}{n_2} } \] Assuming \(S_p^2 = \text{average of } S^2\) (this data is not provided, however, typically \(S_p^2\) might be calculated).
Next, let's assume pooled variance approximation: If we assumed some estimated numbers (this step would require actual data). Let's assume \(s^2 = 1.0\) (if you had data, calculate your true \(S_p^2\)): \[ SE \approx \sqrt{ \frac{1}{4} + \frac{1}{6} } \approx \sqrt{0.25 + 0.16667} \approx \sqrt{0.41667} \approx 0.645 \]
- Finding \(t_{\alpha/2}\) for \(n_1+n_2-2=8\) (at a 99% level):
- \(t_{0.005, 8} \approx 3.355\)
Now we can compute: \[ CI = 2.175 \pm 3.355 \times 0.645 \]
Calculating results: \[ CI = 2.175 \pm 2.157 \approx (0.018, 4.332) \]
Thus the 99% Confidence Interval for the difference between the two treatment means is approximately \( (0.018, 4.332) \).
Note: Actual standard deviations for SE calculation are needed for precise confidence intervals.