R² and Error Variance in Multiple Linear Regression | UPSC ISS 2017 Paper-2 | Problem-5 | RitwikMath

In this video, we solve a **UPSC Statistics / ISS exam PYQ** from **Multiple Linear Regression** involving **coefficient of determination (R²)** and **estimated error variance**.

### Given
- Number of regressors \(k = 5\)
- Number of observations \(n = 25\)
- Sum of Squares due to Regression (SSR) = **360**
- Sum of Squares due to Error (SSE) = **128**

### Step 1: Total Sum of Squares

\[
SST = SSR + SSE
\]

\[
SST = 360 + 128 = 488
\]

### Step 2: Coefficient of Determination

\[
R^2 = \frac{SSR}{SST}
\]

\[
R^2 = \frac{360}{488} \approx 0.7377
\]

So approximately **73.77% ≈ 74% of the total variation is explained by the regressors.**

✅ **Statement 1 is correct.**

### Step 3: Estimated Error Variance

The estimator of error variance is

\[
\hat{\sigma}^2 = \frac{SSE}{n-k-1}
\]

\[
\hat{\sigma}^2 = \frac{128}{25-5-1}
\]

\[
\hat{\sigma}^2 = \frac{128}{19} \approx 6.7368
\]

Since the statement claims the estimated error variance is **3.2**, it is incorrect.

❌ **Statement 2 is incorrect.**

### Final Conclusion
- Statement 1 → **Correct**
- Statement 2 → **Incorrect**

✅ **Correct Option: A**

This type of question is commonly asked in **UPSC Statistics Optional, ISS, and econometrics-related exams**, especially focusing on **ANOVA in regression and interpretation of R² and variance estimates**.

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