Which statement best explains why a t-distribution is used with small samples?

Which statement best explains why a t-distribution is used with small samples?

• A. It accounts for extra uncertainty in estimating the population variance from small samples
• B. It assumes the population variance is known
• C. It is identical to the normal distribution for all sample sizes
• D. It cannot be used for hypothesis testing

Answer: (A)

When the sample size is small, you don’t know the population variance exactly—you have to estimate it from the data. That extra layer of uncertainty changes the behavior of the sampling distribution of the sample mean. The t-distribution is designed to reflect this by having heavier tails than the normal distribution, which accounts for the greater variability you get when the variance is estimated rather than known. The shape of the t-distribution depends on the degrees of freedom (n−1), so with a very small sample it’s spread out more, and as the sample size grows, it becomes more like the normal distribution because the variance estimate becomes more precise. In practice, this is why we use a t-test for the mean when the population variance is unknown, especially with small samples. If the population variance were known, the normal (z) distribution would be used instead; the statement that the distribution is identical to the normal for all sizes is incorrect; and it is not true that t-distribution cannot be used for hypothesis testing—it is specifically used for that purpose.

When the sample size is small, you don’t know the population variance exactly—you have to estimate it from the data. That extra layer of uncertainty changes the behavior of the sampling distribution of the sample mean. The t-distribution is designed to reflect this by having heavier tails than the normal distribution, which accounts for the greater variability you get when the variance is estimated rather than known. The shape of the t-distribution depends on the degrees of freedom (n−1), so with a very small sample it’s spread out more, and as the sample size grows, it becomes more like the normal distribution because the variance estimate becomes more precise.

In practice, this is why we use a t-test for the mean when the population variance is unknown, especially with small samples. If the population variance were known, the normal (z) distribution would be used instead; the statement that the distribution is identical to the normal for all sizes is incorrect; and it is not true that t-distribution cannot be used for hypothesis testing—it is specifically used for that purpose.