Final answer:
The student's question involves comparing areas under a t-distribution to those under a standard normal distribution. As the degrees of freedom increase, the t-distribution more closely resembles the normal distribution, while the tails of both distributions play a key role in determining probabilities.
Step-by-step explanation:
The subject in question is Mathematics, specifically involving probability distributions such as the t-distribution and the standard normal distribution.
In comparing areas under these curves, it's important to understand that the t-distribution accounts for the uncertainty of estimating the population standard deviation with the sample's standard deviation, especially relevant when sample sizes are small (fewer degrees of freedom).
As the number of degrees of freedom increases in a t-distribution, it approaches the shape of the standard normal distribution.
Therefore, the area to the left of a given value for a t-distribution with many degrees of freedom would be similar to that of the standard normal curve.
To find the area under these curves, one would typically use a z-table or t-table (or a software calculator) based on the given z-scores or t-scores and degrees of freedom.
Comparing areas to the right of certain values requires an understanding of the tails of the distributions. The t-distribution has fatter tails than the normal distribution, meaning it gives more probability to extreme values.
This has implications when comparing the areas to the right of the same value on different t-distributions with varying degrees of freedom.