To solve the given problem based on the regression equation with 30 observations:

Step 1: Compute $R^2$ (to 3 decimals)

The formula for $R^2$ is:

$R^2 = \frac{\text{SSR}}{\text{SST}}$

Given:

• SST = 1,802
• SSR = 1,764

$R^2 = \frac{1,764}{1,802} \approx 0.979$

So, $R^2 = 0.979$.

Step 2: Compute $R^2_a$ (to 3 decimals)

The formula for the adjusted $R^2$ is:

$R^2_a = 1 - \left( \frac{(1 - R^2)(n - 1)}{n - k - 1} \right)$

Where:

• $n$ = 30 (number of observations)
• $k$ = number of predictors (not given, assume $k = 1$ if unspecified)
• $R^2 = 0.979$

$R^2_a = 1 - \left( \frac{(1 - 0.979)(30 - 1)}{30 - 1 - 1} \right)$

$R^2_a = 1 - \left( \frac{0.021 \times 29}{28} \right)$

$R^2_a = 1 - 0.02175 \approx 0.978$

So, $R^2_a = 0.978$.

Step 3: Comment on the goodness of fit

Since $R^2 = 0.979$ and $R^2_a = 0.978$, both values are very close to 1. This indicates that the estimated regression equation provides an excellent fit to the data.

If you have further questions, feel free to ask! Here are some related queries you may want to explore:

1. How is adjusted $R^2$ different from $R^2$?
2. Why do we need adjusted $R^2$?
3. How does the number of predictors ($k$) affect adjusted $R^2$?
4. What are the limitations of $R^2$ and $R^2_a$?
5. How can we interpret $R^2$ values for a regression analysis?

Tip: Always check the adjusted $R^2$ when comparing models with different numbers of predictors, as it accounts for model complexity.