Find the value of t for a t -distribution with 22 degrees of freedom such that the area to the left of −t plus the area to the right of t equals 0.02 . Round your answer to three decimal places, if necessary.

To solve this, we need to find the critical value \( t \) for a t-distribution with 22 degrees of freedom, where the combined area in the tails (left of \( -t \) and right of \( t \)) equals 0.02.

### Steps:
1. Since the combined tail area is 0.02, each tail has an area of \( \frac{0.02}{2} = 0.01 \).
2. We look up the \( t \)-value for the upper tail area of 0.01 with 22 degrees of freedom.

Using a t-table or calculator for \( t_{0.01, 22} \):
\[
t \approx 2.508
\]

Thus, the value of \( t \) is approximately:
\[
t = 2.508
\]

### Answer:
The value of \( t \) is \( \boxed{2.508} \).

Would you like further details, or do you have any questions?

### Related Questions:
1. How would you find the \( t \)-value if the degrees of freedom were different?
2. What is the impact of increasing degrees of freedom on the critical \( t \)-value?
3. How can this method be applied in confidence interval estimation?
4. What is the difference between \( t \)-distribution and \( z \)-distribution in hypothesis testing?
5. Why is symmetry important in the t-distribution for this type of calculation?

**Tip:** For two-tailed tests in a t-distribution, always divide the significance level by two to find the critical value in each tail.

Find the value of t for a t-distribution with 22 degrees of freedom such that the area to the left of −t plus the area to the right of t equals 0.02. Round your answer to three decimal places, if necessary.

Learn how to find the critical t-value for a t-distribution with 22 degrees of freedom, given that the combined area in both tails equals 0.02. This solution includes step-by-step guidance and essential statistical concepts, making it suitable for undergraduate statistics students.

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