To solve this problem, you are asked to find the probability $P(T > 2.42)$ for a t-distribution with 12 degrees of freedom.

Solution:

For a t-distribution, we use statistical tables or software (such as a t-distribution calculator) to find the cumulative probability for $T > 2.42$ with 12 degrees of freedom.

1. Using a t-distribution table or calculator, find the cumulative probability up to $T = 2.42$ (this will give $P(T \leq 2.42)$).
2. Subtract this cumulative probability from 1 to get $P(T > 2.42)$.

Step-by-Step:

1. Look up the t-distribution value for $T = 2.42$ and 12 degrees of freedom.
2. The cumulative probability for $T \leq 2.42$ with 12 degrees of freedom is approximately 0.975.
3. So, $P(T > 2.42) = 1 - 0.975 = 0.025$.

Thus, $P(T > 2.42) \approx 0.025$.

Let me know if you need any further clarifications or details. Would you like me to provide more insights into t-distributions or related concepts?

Here are 5 related questions you might find useful:

1. How do we calculate probabilities for t-distributions using software tools like Excel or R?
2. What is the difference between a t-distribution and a normal distribution?
3. How does the number of degrees of freedom affect the shape of a t-distribution?
4. Why is the t-distribution used instead of the normal distribution in certain cases?
5. Can you explain how to construct confidence intervals using the t-distribution?

Tip: The t-distribution becomes more like the standard normal distribution as the degrees of freedom increase.