Explain why the t-distribution has less spread as the number of degrees of freedom increases. Choose the correct answer below. A. The t-distribution has less spread as the degrees of freedom increase because, as $n$ increases, less information is known about $\sigma$ by the law of large numbers. B. The t-distribution has less spread as the degrees of freedom increase because the variability introduced into the t-statistic becomes greater as $\mathrm{n}$ increases. C. The t-distribution has less spread as the degrees of freedom increase because, for large values of $n, n \geq 30$, the t-distribution and the normal distribution are the same. D. The t-distribution has less spread as the degrees of freedom increase because, as $n$ increases, $s$ becomes closer to $\sigma$ by the law of large numbers.

Step 1 ：The t-distribution is a type of probability distribution that is symmetric and bell-shaped, like the standard normal distribution, but has heavier tails, meaning it is more prone to producing values that fall far from its mean. The t-distribution is used in hypothesis testing and in constructing confidence intervals when the sample size is small and/or when the population standard deviation is unknown.

Step 2 ：The degrees of freedom of a t-distribution is related to the sample size. As the degrees of freedom increase, the t-distribution becomes more like a standard normal distribution. This is because as the sample size increases, the sample mean becomes a more accurate estimate of the population mean, and the sample standard deviation becomes a more accurate estimate of the population standard deviation.

Step 3 ：This is a consequence of the law of large numbers, which states that as the size of a sample increases, the sample mean and sample standard deviation converge to the population mean and population standard deviation, respectively.

Step 4 ：Therefore, the correct answer is D. The t-distribution has less spread as the degrees of freedom increase because, as $n$ increases, $s$ becomes closer to $\sigma$ by the law of large numbers.

Step 5 ：Final Answer: \(\boxed{\text{D}}\)