Between the z-distribution and the t-distribution, which statement is true as the degrees of freedom increase?

Between the z-distribution and the t-distribution, which statement is true as the degrees of freedom increase?

• A. The t-distribution has heavier tails than the z-distribution for all degrees of freedom.
• B. As the degrees of freedom increase, the t-distribution approaches the z-distribution.
• C. The z-distribution has heavier tails for large degrees of freedom.
• D. They are identical for all degrees of freedom.

Answer: (B)

The key idea is that the t-distribution accounts for extra uncertainty from estimating the population standard deviation with a sample. When the sample size is small (low degrees of freedom), this extra uncertainty makes the tails heavier compared with the standard normal (z) distribution. As the degrees of freedom increase (larger sample sizes), that extra uncertainty shrinks, and the t-distribution becomes more like the z-distribution. In the limit, when degrees of freedom go to infinity, they are essentially the same. So the statement that as the degrees of freedom increase, the t-distribution approaches the z-distribution is the correct one. The other ideas aren’t true: the z distribution doesn’t gain heavier tails for large df, the two distributions aren’t identical for all degrees of freedom, and the t-distribution does not remain heavier than the z distribution across all df.

The key idea is that the t-distribution accounts for extra uncertainty from estimating the population standard deviation with a sample. When the sample size is small (low degrees of freedom), this extra uncertainty makes the tails heavier compared with the standard normal (z) distribution. As the degrees of freedom increase (larger sample sizes), that extra uncertainty shrinks, and the t-distribution becomes more like the z-distribution. In the limit, when degrees of freedom go to infinity, they are essentially the same.

So the statement that as the degrees of freedom increase, the t-distribution approaches the z-distribution is the correct one. The other ideas aren’t true: the z distribution doesn’t gain heavier tails for large df, the two distributions aren’t identical for all degrees of freedom, and the t-distribution does not remain heavier than the z distribution across all df.