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To solve for the critical values in each case, let's consider the hypothesis testing requirements, where the type of test (z or t) is determined by the sample size and whether the population standard deviation is known. Here's the approach for each scenario:

### Problem Breakdown

1. **Determine Test Type**: Use a t-test when the sample size is small (\( n < 30 \)) and the population standard deviation is unknown. Use a z-test otherwise.

2. **Determine Critical Values**: Based on the significance level (\( \alpha \)) and whether it is a one-tailed or two-tailed test.

Let's go through each part:

### Part (a)
- **Hypothesis**: \( H_0: \mu = 110 \) vs. \( H_A: \mu \neq 110 \)
- **Test**: Since \( n = 41 \) (large sample), use the z-test.
- **Significance Level**: \( \alpha = 0.05 \) (two-tailed)
- **Critical Values**: For a two-tailed test at \( \alpha = 0.05 \), the critical z-values are approximately \( \pm 1.96 \).

### Part (b)
- **Hypothesis**: \( H_0: p = 0.12 \) vs. \( H_A: p > 0.12 \)
- **Test**: This is a test for a proportion, so use the z-test.
- **Significance Level**: \( \alpha = 0.01 \) (one-tailed)
- **Critical Value**: For a one-tailed test at \( \alpha = 0.01 \), the critical z-value is approximately \( 2.33 \).

### Part (c)
- **Hypothesis**: \( H_0: p = 0.8 \) vs. \( H_A: p \neq 0.8 \)
- **Test**: This is a test for a proportion, so use the z-test.
- **Significance Level**: \( \alpha = 0.10 \) (two-tailed)
- **Critical Values**: For a two-tailed test at \( \alpha = 0.10 \), the critical z-values are approximately \( \pm 1.645 \).

### Part (d)
- **Hypothesis**: \( H_0: p = 0.7 \) vs. \( H_A: p < 0.7 \)
- **Test**: This is a test for a proportion, and \( n = 550 \) (large sample), so use the z-test.
- **Significance Level**: \( \alpha = 0.10 \) (one-tailed)
- **Critical Value**: For a one-tailed test at \( \alpha = 0.10 \), the critical z-value is approximately \( -1.28 \).

### Part (e)
- **Hypothesis**: \( H_0: p = 0.5 \) vs. \( H_A: p < 0.5 \)
- **Test**: This is a test for a proportion, so use the z-test.
- **Significance Level**: \( \alpha = 0.10 \) (one-tailed)
- **Critical Value**: For a one-tailed test at \( \alpha = 0.10 \), the critical z-value is approximately \( -1.28 \).

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Do you want a more detailed explanation of any of these steps? Here are some related questions for further exploration:

1. How do you determine whether to use a z-test or a t-test?
2. What is the difference between a one-tailed and a two-tailed test?
3. How do critical values change with different significance levels?
4. How do sample size and population standard deviation affect hypothesis testing?
5. What are some common critical z-values and t-values for different confidence levels?

**Tip**: Always check if the sample size is large enough to justify using the z-test for proportions or means, especially when the population standard deviation is unknown.

For each of the following situations, find the critical value(s) for z or t:

a) H0: μ = 110 vs. HA: μ ≠ 110 at α = 0.05; n = 41
b) H0: p = 0.12 vs. HA: p > 0.12 at α = 0.01
c) H0: p = 0.8 vs. HA: p ≠ 0.8 at α = 0.10
d) H0: p = 0.7 vs. HA: p < 0.7 at α = 0.10; n = 550
e) H0: p = 0.5 vs. HA: p < 0.5 at α = 0.10

Explore how to find critical values for Z or T in hypothesis testing situations, including tests for mean and proportion at different significance levels. Each example includes sample size considerations and the decision between one-tailed and two-tailed tests.

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