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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.

Consider a t-distribution with 12 degrees of freedom. Find P(T > 2.42).

Learn how to calculate the probability P(T > 2.42) for a t-distribution with 12 degrees of freedom. This solution involves using statistical tables or software and key concepts such as the t-distribution and degrees of freedom. Suitable for advanced high school or college-level statistics.

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