Let's solve the given problem step-by-step.

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1. Degrees of Freedom for the F-Test

The degrees of freedom (d.f.) for the F-test are:

• Numerator d.f. (Regression) = $4$ (number of predictors or regressors).
• Denominator d.f. (Error) = $45$ (residual or error degrees of freedom).

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2. Critical Value of F

Using Appendix F for $\alpha = 0.05$, $\text{d.f. numerator} = 4$, and $\text{d.f. denominator} = 45$, the critical value is approximately:

$F_{\text{critical}}(4, 45, 0.05) \approx 2.58.$

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3. F-Statistic Calculation

The F-statistic is calculated using the formula:

$F = \frac{\text{MS}_{\text{Regression}}}{\text{MS}_{\text{Error}}}$

From the table:

• $\text{MS}_{\text{Regression}} = 64,853$
• $\text{MS}_{\text{Error}} = 4,990$

Substitute these values:

$F = \frac{64,853}{4,990} = 13.00$

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4. Is the Regression Significant Overall?

• $F_{\text{calculated}} = 13.00$ exceeds $F_{\text{critical}} = 2.58$.
• This means the regression is significant overall at $\alpha = 0.05$.

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5. $R^2$ and $R^2_{\text{adj}}$ Calculations

(a) $R^2$ Calculation

The formula for $R^2$ is:

$R^2 = \frac{\text{SS}_{\text{Regression}}}{\text{SS}_{\text{Total}}}$

Substitute the values:

• $\text{SS}_{\text{Regression}} = 259,412$
• $\text{SS}_{\text{Total}} = 483,951$

$R^2 = \frac{259,412}{483,951} \approx 0.536 \quad (\text{or } 53.6\%)$

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(b) $R^2_{\text{adj}}$ Calculation

The formula for $R^2_{\text{adj}}$ is:

$R^2_{\text{adj}} = 1 - \left( \frac{1 - R^2}{\frac{n - 1}{n - k - 1}} \right)$

Where:

• $n = 50$ (total observations),
• $k = 4$ (number of predictors),
• $R^2 = 0.536$.

Substitute these values:

$R^2_{\text{adj}} = 1 - \left( \frac{1 - 0.536}{\frac{50 - 1}{50 - 4 - 1}} \right)$

Simplify:

= 1 - (0.464 \times 0.918) \approx 1 - 0.426 = 0.574 \quad (\text{or } 57.4\%)$$ --- ### **Summary of Results** 1. **Degrees of Freedom:** $$\text{Regression} = 4$$, $$\text{Error} = 45$$. 2. **Critical $$F$$:** $$2.58$$. 3. **Calculated $$F$$:** $$13.00$$. 4. **Significance:** Regression is significant overall. 5. **$$R^2$$:** $$0.536$$ (53.6%). 6. **$$R^2_{\text{adj}}$$:** $$0.574$$ (57.4%). --- Do you want more details or have any questions? ### **Related Questions** 1. What happens to $$R^2_{\text{adj}}$$ if more predictors are added unnecessarily? 2. How does $$\alpha$$ influence the critical value of $$F$$? 3. What are the assumptions for conducting an F-test in regression? 4. How does $$R^2$$ interpret the proportion of variance explained? 5. When is $$R^2_{\text{adj}}$$ preferred over $$R^2$$? ### **Tip** Always verify the assumptions of linear regression (normality, independence, homoscedasticity, and linearity) before interpreting $$F$$-test results.