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. True
• C. Sometimes
• D. Not sure

Answer: (B)

As you collect more data, the uncertainty from estimating the population standard deviation decreases, so the t statistic behaves more like a standard normal. The t distribution uses s in the denominator, and with larger n (degrees of freedom growing), s becomes a more accurate substitute for the true sigma. This causes the heavier tails of the t distribution to fade and the shape to converge toward N(0,1). In practice, with moderately large samples (around 30 or more), the t distribution is already very close to the standard normal. So the statement is true. The other options don’t fit because the convergence to the normal isn’t random or limited to some cases—it becomes increasingly similar as sample size increases.

As you collect more data, the uncertainty from estimating the population standard deviation decreases, so the t statistic behaves more like a standard normal. The t distribution uses s in the denominator, and with larger n (degrees of freedom growing), s becomes a more accurate substitute for the true sigma. This causes the heavier tails of the t distribution to fade and the shape to converge toward N(0,1). In practice, with moderately large samples (around 30 or more), the t distribution is already very close to the standard normal. So the statement is true. The other options don’t fit because the convergence to the normal isn’t random or limited to some cases—it becomes increasingly similar as sample size increases.