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Let's go through the three parts of your homework related to finding the P-values for various test statistics.

### A) Test Statistic \( t_2 = -2.08 \)
1. **Left-tailed P-Value**: Since the test statistic is negative, this will be relevant for a left-tailed test. To find the left-tailed P-value, you can use a t-distribution calculator or a graphing calculator with the `tcdf` function.
   \[
   \text{Left-tailed P-Value} = P(T \leq -2.08) \text{ for appropriate degrees of freedom (df)}.
   \]

2. **Right-tailed P-Value**: This is the complement of the left-tailed P-value for this negative statistic. Use:
   \[
   \text{Right-tailed P-Value} = 1 - P(T \leq -2.08).
   \]

3. **Two-tailed P-Value**: Double the left-tailed P-value:
   \[
   \text{Two-tailed P-Value} = 2 \times P(T \leq -2.08).
   \]

### B) Test Statistic \( Z = 1.77 \)
1. **Left-tailed P-Value**: Find the area under the standard normal curve to the left of \( Z = 1.77 \). Use a Z-table or a standard normal distribution calculator.
   \[
   P(Z \leq 1.77).
   \]

2. **Right-tailed P-Value**: This is simply the area to the right:
   \[
   \text{Right-tailed P-Value} = 1 - P(Z \leq 1.77).
   \]

3. **Two-tailed P-Value**: Since it's a two-tailed test, you will need to double the smaller of the left- or right-tailed P-values:
   \[
   \text{Two-tailed P-Value} = 2 \times P(Z \geq 1.77).
   \]

### C) Test Statistic \( \chi^2_1 = 5.8 \)
1. **Left-tailed P-Value**: For a chi-squared test, P-values are typically found using the right tail, as the chi-squared distribution is not symmetric. But if you were looking for a left-tailed P-value, you could still find it using a chi-square table or a calculator with degrees of freedom \( df = 1 \).

2. **Right-tailed P-Value**: Find the area to the right of the test statistic:
   \[
   \text{Right-tailed P-Value} = P(\chi^2 \geq 5.8 \text{ with df = 1}).
   \]

3. **Two-tailed P-Value**: Since the chi-squared distribution is typically used for right-tailed tests, there isn't a conventional two-tailed test for chi-squared.

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Would you like me to walk through any particular calculations in more detail?

Here are some related questions you might consider:
1. How can you calculate the degrees of freedom in each of these tests?
2. What is the difference between Z-tests and t-tests in hypothesis testing?
3. When would you use a one-tailed vs. a two-tailed test?
4. What significance level (\( \alpha \)) are you using for these tests?
5. How does the shape of the chi-square distribution change with degrees of freedom?

**Tip:** Always check whether your test statistic uses a Z-distribution, t-distribution, or chi-square distribution to ensure you're using the correct table or function.

For each of these problems, neatly draw the appropriate graph with all of the relevant details, write down exactly which calculator function you used to find your answer, and the final answer itself. Round all answers to 4 decimal places.

(A) The test statistic is t2 = -2.08.
(B) The test statistic is Z = 1.77.
(C) The test statistic is χ²₁ = 5.8.

This solution covers hypothesis testing and P-value calculations for different test statistics, including t = -2.08, Z = 1.77, and χ²₁ = 5.8. Learn how to find left-tailed, right-tailed, and two-tailed P-values using appropriate distribution tables and calculator functions. Ideal for college-level statistics students.

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