The t-statistic is calculated by dividing:

Question

BlackTom AI · Accepted Answer

The t-statistic equals the estimator minus its hypothesized value divided by the standard error of the estimator.

1. Relevant knowledge points and concepts
- Estimator (e.g., β̂) is the sample estimate of a population parameter (e.g., slope β).
- Null hypothesis H0 specifies a hypothesized value β0 (often 0).
- Standard error (SE(β̂)) measures the estimated standard deviation of the sampling distribution of the estimator.
- The t-statistic measures how many standard errors the estimator is away from the hypothesized value, and it follows a Student’s t-distribution under H0 (with n − k degrees of freedom in regression).

2. Complete reasoning process step by step
- Under H0: β = β0, compute the difference between estimate and hypothesized value: β̂ − β0.
- Scale that difference by the estimated standard deviation of β̂, i.e., divide by SE(β̂).
- The resulting ratio is the t-statistic: t = (β̂ − β0) / SE(β̂).
- This standardized quantity allows comparison to critical values from the t-distribution to decide whether to reject H0.

3. Example calculation with intermediate steps
- Suppose β̂ = 2.5, hypothesized β0 = 0, and SE(β̂) = 0.5.
- Compute numerator: 2.5 − 0 = 2.5.
- Divide by SE: 2.5 / 0.5 = 5.0.
- So t = 5.0; compare to t-critical (e.g., ≈2 for typical α = 0.05) and reject H0.

4. Why the correct answer is correct
- The t-statistic is by definition the standardized difference between estimate and hypothesized value. Dividing by the standard error converts the raw difference into units of sampling variability, enabling probabilistic inference via the t-distribution.

5. Why the other options are incorrect
- "the slope plus or minus twice the standard error": describes an approximate 95% confidence interval, not the t-statistic.
- "the OLS estimator by twice its standard error": arbitrarily divides by 2·SE; the correct denominator is SE, not twice SE.
- "the slope by the standard deviation of the explanatory variable": unrelated; this would not standardize the estimator by its sampling variability and is not the t-statistic.

6. Conclusion
The t-statistic is (estimator − hypothesized value) divided by the estimator’s standard error, which matches the provided correct answer.

The t-statistic is calculated by dividing:单项选择题

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