## Related questions with answers

Based on a sample of 30 observations, the population regression model

$y_i=\beta_0+\beta_1 x_i+\varepsilon_i$

was estimated. The least squares estimates obtained were as follows:

$b_0=10.1 \text { and } b_1=8.4$

The regression and error sums of squares were as follows:

$S S R=128 \text { and } \quad S S E=286$

a. Find and interpret the coefficient of determination.

b. Test at the $10 \%$ significance level against a twosided alternative the null hypothesis that $\beta_1$ is 0 .

c. Find

$\sum_{i=1}^{30}\left(x_i-\bar{x}\right)^2$

Solution

Verified**(a)** The coefficient of determination, denoted as $R^2$, can be calculated using the formula:

$R^2=1-\frac{SSE}{SST}$

where $SSE$ is the sum of squares error and $SST$ is the sum of squares total. Since the $SST$ is unknown, we must calculate such value first to determine the coefficient of determination.

Calculate the $SST$ by using the formula below.

$SST=SSR+SSE$

Plug in the values $SSR=128$ and $SSE=286$ to the formula above.

$SST=128+286$

Simplify by addition.

$SST=414$

The $SST$ is equal to $414$.

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