(For this problem, let β1hat be the OLS estimate of β1.) The 95% confidence interval for β1 is the interval:

Question

(For this problem, let β1hat be the OLS estimate of β1.)  The 95% confidence interval for β1 is the interval:

BlackTom AI · Accepted Answer

The sampling distribution of the OLS slope estimator β1hat is approximately normal with mean β1 and standard error SE(β1hat), so a two-sided 95% confidence interval is constructed as estimator ± critical value × standard error.

1. Relevant concepts
- β1hat is the OLS estimator of the true slope β1.
- SE(β1hat) = sqrt(VAR(β1hat)) measures the estimator’s sampling variability.
- For a 95% two-sided interval, the critical value from the standard normal is 1.96 (or from the t-distribution with appropriate df for small samples).
- The CI aims to produce an interval that contains the true parameter β1 with 95% probability in repeated samples.

2. Step-by-step reasoning
Start with the standardized variable:
(β1hat − β1) / SE(β1hat) ≈ N(0,1).
For 95% central probability,
P(−1.96 < (β1hat − β1) / SE(β1hat) < 1.96) ≈ 0.95.

3. Algebra / intermediate steps
Multiply through by SE(β1hat):
P(−1.96 SE(β1hat) < β1hat − β1 < 1.96 SE(β1hat)) ≈ 0.95.
Rearrange to isolate β1:
P(β1hat − 1.96 SE(β1hat) < β1 < β1hat + 1.96 SE(β1hat)) ≈ 0.95.
Thus the 95% CI is (β1hat − 1.96 SE(β1hat), β1hat + 1.96 SE(β1hat)).

4. Why the correct answer is correct
This interval directly follows from the approximate normal sampling distribution of β1hat and uses the 1.96 critical value appropriate for 95% two-sided confidence. The lower bound is the estimator minus 1.96 times its standard error; the upper bound is the estimator plus 1.96 times its standard error.

5. Why other options are incorrect
- "(β1 -96SE(β1), β1 + 1.96SE(β1)).": Uses the true parameter β1 in the endpoints (not the estimator) and has a typo “-96”; CI must be centered on β1hat.
- "(β1hat -96, β1hat + 1.96).": Drops the SE factor and has a typo “-96”; numerical shifts without SE are meaningless.
- "(β1hat -1.645SE)(β1hat),β1hat + 1.645SE(β1hat)).": Uses 1.645 (the 90% z critical value), not 1.96, and is misformatted.

6. Conclusion
The correct 95% confidence interval is (β1hat  -1.96SE(β1hat), β1hat + 1.96SE(β1hat)). This matches the provided answer.

(For this problem, let β1hat be the OLS estimate of β1.) The 95% confidence interval for β1 is the interval:单项选择题

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