True or False: The t-distribution becomes more similar to the standard normal distribution as the sample size increases.

True or False: The t-distribution becomes more similar to the standard normal distribution as the sample size increases.

• A. False
• B. Sometimes
• C. True
• D. Not sure

Answer: (C)

True—the t-distribution becomes more like the standard normal as the sample size increases. The t distribution accounts for the extra uncertainty of estimating the population standard deviation from the sample. With small samples, this uncertainty makes the t shape have heavier tails and be more spread out than the normal distribution. As the sample size grows, the estimate of the standard deviation becomes more accurate and the degrees of freedom increase, causing the tails to thin and the shape to tighten toward the standard normal. In the limit, as degrees of freedom go to infinity, the t distribution converges to the standard normal.

True—the t-distribution becomes more like the standard normal as the sample size increases. The t distribution accounts for the extra uncertainty of estimating the population standard deviation from the sample. With small samples, this uncertainty makes the t shape have heavier tails and be more spread out than the normal distribution. As the sample size grows, the estimate of the standard deviation becomes more accurate and the degrees of freedom increase, causing the tails to thin and the shape to tighten toward the standard normal. In the limit, as degrees of freedom go to infinity, the t distribution converges to the standard normal.