Question

which is larger, the area under the t-distribution with 10 degrees of freedom to the right of t2.32 or the area under the standard normal distribution to the right of z2.32? The area under the t-distribution with 10 degrees of freedom to the right of t=2.32 is the area under the standard normal distribution to the right of z=2.32.

Answer 1

The area under the t-distribution with 10 degrees of freedom to the right of t=2.32 is smaller than the area under the standard normal distribution to the right of z=2.32.

Step-by-step explanation:

The area under the t-distribution with 10 degrees of freedom to the right of t=2.32 is smaller than the area under the standard normal distribution to the right of z=2.32.

To find the areas, you can use the t-table and the z-table. Looking at the t-table for the t-distribution with 10 degrees of freedom, the area to the right of t=2.32 is approximately 0.02.

On the other hand, looking at the z-table for the standard normal distribution, the area to the right of z=2.32 is approximately 0.0103. Since 0.02 is greater than 0.0103, the area under the t-distribution is larger.

Answer 2

The area under the t-distribution with 10 degrees of freedom to the right of t=2.32 is larger than the area under the standard normal distribution to the right of z=2.32, because the t-distribution has heavier tails than the z-distribution.

Step-by-step explanation:

The question pertains to comparing the areas under two different probability distributions to the right of a specified value. More specifically, we need to determine which is larger: the area under the t-distribution with 10 degrees of freedom to the right of t=2.32, or the area under the standard normal distribution (also known as the z-distribution) to the right of z=2.32.

The area under the normal curve for any value can be found using a z-table, which gives us the cumulative area to the left of a specific z-score. To find the area to the right, we subtract this value from 1. However, because the t-distribution has heavier tails than the standard normal distribution, for the same t or z value the area to the right (or the p-value) is larger for a t-distribution with finite degrees of freedom compared to the standard normal distribution. Hence the area to the right of 2.32 in the t-distribution with 10 df will be larger than the area to the right of 2.32 in the standard z-distribution.

It's also noteworthy that as the degrees of freedom increase, the t-distribution approaches the standard normal distribution. Therefore, the difference in areas for a given t or z value will diminish as the number of degrees of freedom increases.