Which of the following is true regarding the t-distribution?

Question

BlackTom AI · Accepted Answer

When evaluating statements about the t-distribution, it helps to recall its key properties and when it is used.
Option 1: 'Similar to a chi-square statistic, a t statistic of a hypothesis test cannot be negative.' This is incorrect because the t statistic can be negative, reflecting the direction of the difference or effect relative to the null hypothesis. The sign indicates whether the observed estimate is below or above the hypothesized value, so a negative t value is perfectly possible and common.
Option 2: 'The sample size and the degrees of freedom are inversely related.' This is not accurate. For a one-sample t-test, the degrees of freedom are typically n − 1, which increases as the sample size n increases. Therefore, they are directly related in the sense that larger samples yield larger degrees of freedom, not inversely related.
Option 3: 'The t-distribution can used when performing inferences for the difference of two population means for dependent samples.' This statement aligns with standard practice. When comparing means from paired or dependent samples, we use the paired t-test, which relies on the t-distribution with appropriate degrees of freedom to perform inference about the mean difference.
Option 4: 'With increased sample size, the peak of the distribution becomes shorter and the tails of the distribution become taller.' This describes a distribution that becomes flatter and heavier-tailed as sample size grows, which is the opposite of what happens. In fact, as sample size increases, the t-distribution becomes more like the standard normal: the peak becomes taller (not shorter) and the tails become thinner (not taller).
In summary, the third option correctly states a true use of the t-distribution, while the other options misstate properties of the t-distribution or the behavior of its shape as sample size grows.

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Which of the following is true regarding the t-distribution?

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