Case
Refer to the ANOVA table below. 

Source
d.f
SS
MS
Regression	4	259,412	64,853
Error	45	224,539	4,990
Total	49	483,951	
 

State the degrees of freedom for the F test for the overall significance.
Use Appendix F to look up the critical value of F for a = .05.
Calculate the F statistic.  Is the regression significant overall?
Calculate R2 and R2auj,, Showing your formulas clearly.

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

---

### **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).

---

### **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:

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

---

### **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:

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

---

### **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 \( \alpha = 0.05 \).

---

### **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:
- \( \text{SS}_{\text{Regression}} = 259,412 \)
- \( \text{SS}_{\text{Total}} = 483,951 \)

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

---

#### **(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:
\[
R^2_{\text{adj}} = 1 - \left( \frac{0.464}{\frac{49}{45}} \right)
= 1 - (0.464 \times 0.918) \approx 1 - 0.426 = 0.574 \quad (\text{or } 57.4\%)
\]

---

### **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.

Refer to the ANOVA table below. 

Source
d.f
SS
MS
Regression	4	259,412	64,853
Error	45	224,539	4,990
Total	49	483,951	

State the degrees of freedom for the F test for the overall significance.
Use Appendix F to look up the critical value of F for a = .05.
Calculate the F statistic. Is the regression significant overall?
Calculate R2 and R2adj, showing your formulas clearly.

Analyze an ANOVA table with step-by-step calculations for degrees of freedom, F-statistic, critical value of F, overall significance, R^2, and adjusted R^2. Learn how to determine the significance of regression analysis and understand the formulas involved. Suitable for college or university-level statistics students.

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