Which is larger, the area under the t-distribution with 10 degrees of freedom to the right of t = 2.30 or the area under the standard normal distribution to the right of z = 2.32? Why?

Verified step by step guidance

1. Understand that the problem asks us to compare two tail probabilities: one from a t-distribution with 10 degrees of freedom at t = 2.30, and the other from a standard normal distribution at z = 2.32.
2. Recall that the t-distribution with finite degrees of freedom has heavier tails than the standard normal distribution, meaning it assigns more probability to extreme values farther from zero.
3. Express the areas as probabilities: the area to the right of t = 2.30 is \(P(T > 2.30)\) where \(T \sim t_{10}\), and the area to the right of z = 2.32 is \(P(Z > 2.32)\) where \(Z \sim N(0,1)\).
4. Since the t-distribution has heavier tails, for values near 2.3, \(P(T > 2.30)\) will be larger than \(P(Z > 2.32)\), even though 2.32 is slightly larger than 2.30, because the t-distribution's tail probabilities decrease more slowly.
5. Therefore, the area under the t-distribution to the right of 2.30 is larger than the area under the standard normal distribution to the right of 2.32, due to the heavier tails of the t-distribution with 10 degrees of freedom.

Key Concepts

t-Distribution

The t-distribution is a probability distribution used when estimating population parameters with small sample sizes. It is similar to the normal distribution but has heavier tails, meaning it accounts for more variability. The shape depends on degrees of freedom; fewer degrees of freedom result in wider tails.

Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is symmetric and bell-shaped, used as a reference for z-scores. Probabilities correspond to areas under the curve, with values farther from zero indicating more extreme outcomes.

Comparing Tail Areas (Right-Tail Probabilities)

The area to the right of a given value under a distribution curve represents the probability of observing a value greater than that point. Because the t-distribution has heavier tails, for similar cutoff values, the right-tail area under the t-distribution is larger than under the normal distribution, especially with low degrees of freedom.