Question 1195287
**1. Sum of Squared Errors (SSE)**

* **R-squared (R²)** is the coefficient of determination, which represents the proportion of the total variation in the dependent variable (Y) that is explained by the independent variable (X). 
* **Formula:** R² = 1 - (SSE / SST) 
    * where:
        * SSE: Sum of Squared Errors
        * SST: Total Sum of Squares

* **Rearrange to solve for SSE:**
    * SSE = SST * (1 - R²)
    * SSE = 300 * (1 - 0.5) 
    * SSE = 300 * 0.5 
    * SSE = 150

**2. Adjusted R-squared**

* **Formula:**
    * Adjusted R² = 1 - [(1 - R²) * (n - 1) / (n - k - 1)]
        * where:
            * n: number of observations (22)
            * k: number of independent variables (1 in this case)

* **Calculate:**
    * Adjusted R² = 1 - [(1 - 0.5) * (22 - 1) / (22 - 1 - 1)]
    * Adjusted R² = 1 - (0.5 * 21 / 20)
    * Adjusted R² = 1 - 1.05 
    * Adjusted R² = -0.05 

* **Note:** The adjusted R-squared can be negative in some cases, especially when the model has a poor fit.

**3. Unbiased Estimator of σ²**

* **Mean Squared Error (MSE):** 
    * MSE = SSE / (n - k - 1) 
    * MSE = 150 / (22 - 1 - 1) 
    * MSE = 150 / 20 
    * MSE = 7.5

* **Unbiased Estimator of σ²:** MSE = 7.5

**4. F-statistic**

* **Formula:** F = (Regression Mean Square / Residual Mean Square) 
    * Regression Mean Square (MSR) = SSR / k 
        * SSR (Sum of Squares Regression) = SST * R² = 300 * 0.5 = 150
        * MSR = 150 / 1 = 150
    * Residual Mean Square (MSE) = 7.5 (calculated earlier)

* **F-statistic:** F = 150 / 7.5 = 20

**5. Critical t-value**

* **Hypotheses:**
    * H₀: β₁ = 0 (No significant relationship between X and Y)
    * H₁: β₁ > 0 (X has a significant positive effect on Y) 
* **Significance Level:** α = 0.05
* **Degrees of Freedom:** n - 2 = 22 - 2 = 20
* **One-tailed test:** Since we are testing for a positive effect.

* **Using a t-distribution table or statistical software:** 
    * The critical t-value for a one-tailed test with α = 0.05 and 20 degrees of freedom is approximately 1.725.

**In Summary:**

1. SSE = 150
2. Adjusted R² = -0.05
3. Unbiased Estimator of σ² = 7.5
4. F-statistic = 20
5. Critical t-value ≈ 1.725

I hope this helps! Let me know if you have any other questions.