Question

Consider the simple linear regression model $y_i = \alpha + \beta x_i + \varepsilon_i$, where $\varepsilon_i$ are i.i.d. random variables with mean 0 and variance $\sigma^2$. Given that $n = 20$, $\sum x_i = 100$, $\sum y_i = 50$, $\sum x_i^2 = 600$, $\sum y_i^2 = 500$, and $\sum x_i y_i = 400$, find the least squares estimates of $\alpha$ and $\beta$.

• A. 5 and $\dfrac{3}{2}$
• B. $-5$ and $\dfrac{3}{2}$
• C. 5 and $-\dfrac{3}{2}$
• D. $-5$ and $-\dfrac{3}{2}$

Hint:

Always center data using $\bar{x}, \bar{y}$ when computing regression coefficients — this minimizes computational errors.

Answer 1

The Correct Option is B

Least squares estimates formulas:

$$\hat{\beta} = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}$$

$$\hat{\alpha} = \bar{y} - \hat{\beta}\bar{x}$$

where $\bar{x} = \frac{\sum x_i}{n}$ and $\bar{y} = \frac{\sum y_i}{n}$

Step 1: Calculate means

$$\bar{x} = \frac{100}{20} = 5$$

$$\bar{y} = \frac{50}{20} = 2.5$$

Step 2: Calculate $\hat{\beta}$

$$\hat{\beta} = \frac{20 \times 400 - 100 \times 50}{20 \times 600 - 100^2}$$

$$= \frac{8000 - 5000}{12000 - 10000}$$

$$= \frac{3000}{2000}$$

$$= \frac{3}{2}$$

Step 3: Calculate $\hat{\alpha}$

$$\hat{\alpha} = \bar{y} - \hat{\beta}\bar{x}$$

$$= 2.5 - \frac{3}{2} \times 5$$

$$= 2.5 - 7.5$$

$$= -5$$

Answer: (B) $\alpha = -5$ and $\beta = \frac{3}{2}$