As the number of degrees of freedom for a t distribution increases, what happens to its relationship with the standard normal distribution?

As the number of degrees of freedom for a t distribution increases, it approaches the characteristics of the standard normal distribution. This phenomenon occurs because the t distribution is wider and has heavier tails than the standard normal distribution when the degrees of freedom are low. However, as the degrees of freedom increase, the t distribution converges towards the normal distribution, meaning that the differences in their shapes diminish.

This convergence is important in statistical inference, particularly in scenarios involving hypothesis testing and confidence intervals. With a larger sample size – which is associated with higher degrees of freedom – the sample mean becomes a better estimator for the population mean, leading to results that align more closely with those derived from the standard normal distribution.

Consequently, the correct response reflects that as degrees of freedom increase, the disparity between the t distribution and the standard normal distribution becomes smaller, illustrating that the t distribution becomes more like the standard normal distribution under such conditions.