How does a t-distribution differ from the standard normal distribution?

How does a t-distribution differ from the standard normal distribution?

• A. The t-distribution has lighter tails than the normal.
• B. The t-distribution has thicker tails and converges to the normal as degrees of freedom increase.
• C. The t-distribution is used when the population standard deviation is known and the sample size is large.
• D. The t-distribution has fixed variance regardless of sample size.

Answer: (B)

The key idea is that the t-distribution reflects extra uncertainty from estimating the population standard deviation from the sample. When σ is unknown and we use the sample standard deviation in place of it, the resulting distribution has heavier tails than the standard normal, because there's more variability left in the estimate of scale. As the degrees of freedom increase (which happens when you have a larger sample size), this extra uncertainty diminishes, and the t-distribution becomes increasingly similar to the standard normal, eventually approaching it in the limit. Why the other statements don’t fit: lighter tails would imply less extreme values than the normal, which isn’t true for finite samples—the t-distribution has fatter tails. Using the t-distribution when σ is known and the sample size is large is unnecessary; in that situation you would normally use the standard normal. The variance of the t-distribution isn’t fixed—it depends on the degrees of freedom and is greater than 1 for finite df, approaching 1 as df grows; that dependency on sample size is another hallmark that distinguishes it from the standard normal.

The key idea is that the t-distribution reflects extra uncertainty from estimating the population standard deviation from the sample. When σ is unknown and we use the sample standard deviation in place of it, the resulting distribution has heavier tails than the standard normal, because there's more variability left in the estimate of scale.

As the degrees of freedom increase (which happens when you have a larger sample size), this extra uncertainty diminishes, and the t-distribution becomes increasingly similar to the standard normal, eventually approaching it in the limit.

Why the other statements don’t fit: lighter tails would imply less extreme values than the normal, which isn’t true for finite samples—the t-distribution has fatter tails. Using the t-distribution when σ is known and the sample size is large is unnecessary; in that situation you would normally use the standard normal. The variance of the t-distribution isn’t fixed—it depends on the degrees of freedom and is greater than 1 for finite df, approaching 1 as df grows; that dependency on sample size is another hallmark that distinguishes it from the standard normal.