If the p-value associated with the observed t-statistic for the independent variable in a simple linear regression is less than .05, the critical value of t in the t-distribution:

Question

BlackTom AI · Accepted Answer

The question asks about the relationship between the p-value of the observed t-statistic in a simple linear regression and the corresponding critical value of t.
Option: 'Will be smaller than the observed t-statistic.'
To reason this out, remember that a p-value less than 0.05 means the observed t-statistic falls in the rejection region for a two-tailed test at alpha = 0.05 (or in one-tailed if specified, but commonly two-tailed is assumed). In such cases, the critical t value, t_crit, is the boundary that separates non-rejection from rejection: |t_obs| > t_crit. Therefore, the magnitude of the critical value is indeed smaller than the magnitude of the observed t-statistic.
Let’s contrast with the intuitive alternative: if p < 0.05, you are deeper into the tail than the cutoff, implying the cutoff threshold (the critical value) lies closer to zero than the observed statistic, not larger. This supports the statement that the critical value is smaller in magnitude than the observed t-statistic.
If we imagined a scenario where p were larger than 0.05, the observed t would not exceed the critical boundary, and the inequality |t_obs| > t_crit would not hold; but that is not the current situation. Thus, the claim in the option aligns with the standard decision rule for hypothesis testing using the t-distribution.
In sum, the observed t-statistic surpasses the critical t value in magnitude when the p-value is below 0.05, making the critical value smaller than the observed t-statistic.

If the p-value associated with the observed t-statistic for the independent variable in a simple linear regression is less than .05, the critical value of t in the t-distribution:单项选择题

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