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To solve this, we need to compute \( s^2 \), which is the estimator of the variance \( \sigma^2 \) of the random error term \( \epsilon \). The formula for \( s^2 \) when using a least squares fit is:

\[
s^2 = \frac{SSE}{n - 2}
\]

Where:
- \( SSE \) is the sum of squared errors, which is given as 0.766.
- \( n \) is the number of data points, which is 26.

So, let's compute \( s^2 \):

\[
s^2 = \frac{0.766}{26 - 2}
\]

I will calculate this for you.The value of \( s^2 \), the estimator of \( \sigma^2 \), is approximately 0.0319 when rounded to four decimal places.

Would you like to proceed with part (b), which asks for the estimate of \( \sigma \), or clarify any details?

Here are 5 questions to deepen your understanding:
1. How is the sum of squared errors (SSE) computed in a least squares regression?
2. Why do we divide by \( n - 2 \) instead of \( n \) when estimating \( s^2 \)?
3. How does increasing the number of data points affect the value of \( s^2 \)?
4. Can \( s^2 \) be used to directly estimate the standard error of the regression?
5. What are the assumptions about errors when performing least squares fitting?

**Tip:** The variance \( \sigma^2 \) describes the spread of the random error term, while \( \sigma \) (its square root) represents the standard deviation of these errors.

Suppose you fit a least squares line to 26 data points and the calculated value of SSE is 0.766. a. Find s^2, the estimator of σ^2 (the variance of the random error term ε). b. Find s, the estimate of σ. c. What is the largest deviation that you might expect between any one of the 26 points and the least squares line?

Learn how to calculate the variance and standard deviation in a least squares regression problem with 26 data points, using the sum of squared errors (SSE = 0.766). This solution includes key concepts such as variance estimation, the least squares method, and calculating the largest deviation from the regression line.

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