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.

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.