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: Recall the OLS estimators.
For the simple linear regression model
\[ Y_i=\alpha+\beta X_i+u_i \] the ordinary least squares estimators are
\[ \hat{\beta} = \frac{\sum X_iY_i-\frac{\sum X_i\sum Y_i}{n}} {\sum X_i^2-\frac{(\sum X_i)^2}{n}} \] and
\[ \hat{\alpha} = \bar{Y}-\hat{\beta}\bar{X} \]
where
\[ \bar{X}=\frac{\sum X_i}{n} \] and
\[ \bar{Y}=\frac{\sum Y_i}{n} \]

Step 2: Calculate the sample means.
Given,
\[ n=100 \] \[ \sum X_i=6000 \] \[ \sum Y_i=500 \]
Therefore,
\[ \bar{X}=\frac{6000}{100}=60 \] and
\[ \bar{Y}=\frac{500}{100}=5 \]

Step 3: Calculate the slope coefficient \(\hat{\beta}\).
Using the formula,
\[ \hat{\beta} = \frac{40000-\frac{(6000)(500)}{100}} {460000-\frac{(6000)^2}{100}} \]
First calculate the numerator:
\[ \frac{(6000)(500)}{100} = 30000 \] Thus,
\[ 40000-30000=10000 \]
Now calculate the denominator:
\[ \frac{(6000)^2}{100} = \frac{36000000}{100} = 360000 \] Therefore,
\[ 460000-360000=100000 \]
Hence,
\[ \hat{\beta} = \frac{10000}{100000} = 0.1 \]

Step 4: Calculate the intercept estimator \(\hat{\alpha}\).
Use the formula
\[ \hat{\alpha} = \bar{Y}-\hat{\beta}\bar{X} \]
Substitute the values:
\[ \hat{\alpha} = 5-(0.1)(60) \]
\[ \hat{\alpha} = 5-6 \] \[ \hat{\alpha} = -1 \]

Step 5: Verify the calculation.
The estimated regression equation becomes
\[ \hat{Y} = -1+0.1X \]
Substituting
\[ X=60 \] gives
\[ \hat{Y} = -1+0.1(60) \] \[ \hat{Y} = -1+6 = 5 \]
which matches the sample mean of \(Y\). Hence, the calculation is correct.

Step 6: Final conclusion.
Therefore, the estimated value of
\[ \alpha \] is
\[ \boxed{-1} \]