Question 1195287: A manager wants to predict the cost (Y) of travel for salespeople based on the number of days (X) spent on each sales trip. Using 22 observations the linear equation Y = 40 + 12X is fitted using Ordinary Least Squares (OLS) technique and the R2 value is 0.5.
Required;
1. What is the sum of all squared errors of regression?
2. What is the value of the Adjusted R2 statistic?
3. Given that the Total Sum of Squares (SST) is 300, what is the value of an unbiased estimator of σ2, the variance of the random term?
4. What is the value of the F-statistic used to test for the overall relevance of the model?
5. The manager wishes to test (at 5% level of significance) the hypothesis that X has a significant positive effect on Y. Which of the following best approximates the critical t-value for the test?
please help me solve this question
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **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.
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