Question

Consider the following population regression equation:

Hint:

In simple linear regression, first calculate the slope coefficient using covariance and variance terms, then use \(\hat{\alpha}=\bar{Y}-\hat{\beta}\bar{X}\).

Answer 1

Correct Answer: -1

Step 1: Find the means.
With $n=100$, $\sum X=6000$ and $\sum Y=500$, the means are $\bar{X}=60$ and $\bar{Y}=5$.

Step 2: Compute the slope.
\[ \hat{\beta}=\frac{\sum XY-\dfrac{\sum X\sum Y}{n}}{\sum X^2-\dfrac{(\sum X)^2}{n}}=\frac{40000-30000}{460000-360000}=\frac{10000}{100000}=0.1 \]

Step 3: Compute the intercept.
\[ \hat{\alpha}=\bar{Y}-\hat{\beta}\bar{X}=5-0.1(60)=5-6=-1 \]

Step 4: Quick check.
The line $\hat{Y}=-1+0.1X$ at $X=60$ gives $5$, which is the mean of $Y$. Good.
\[ \boxed{-1} \]