As the sample size grows large, the t-distribution becomes more like the standard normal distribution.

As the sample size grows large, the t-distribution becomes more like the standard normal distribution.

• A. Yes
• B. No, it remains different from the standard normal
• C. It becomes uniform
• D. It becomes skewed

Answer: (B)

As the degrees of freedom grow, the t-distribution becomes more like the standard normal because the extra uncertainty from estimating the standard deviation shrinks. The t-distribution starts with heavier tails when sample sizes are small, reflecting that S, the sample standard deviation, is a noisy estimate of the true sigma. As n increases, S becomes a more accurate estimate of sigma, and the t-statistic (which standardizes the sample mean using S) behaves more like a standard normal variable. In the limit of large n (or large degrees of freedom), the t-distribution converges to the standard normal distribution N(0,1). So the statement is true.

As the degrees of freedom grow, the t-distribution becomes more like the standard normal because the extra uncertainty from estimating the standard deviation shrinks. The t-distribution starts with heavier tails when sample sizes are small, reflecting that S, the sample standard deviation, is a noisy estimate of the true sigma. As n increases, S becomes a more accurate estimate of sigma, and the t-statistic (which standardizes the sample mean using S) behaves more like a standard normal variable. In the limit of large n (or large degrees of freedom), the t-distribution converges to the standard normal distribution N(0,1). So the statement is true.