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Let's break down the information given and solve the questions one by one.

### Problem Overview
1. **Hypotheses**:
   - Null Hypothesis (\(H_0\)): \( p = 0.5 \)
   - Alternative Hypothesis (\(H_1\)): \( p \neq 0.5 \)

2. **Significance Level**: \( \alpha = 0.10 \) (indicating a 10% significance level)

3. **Test Statistic**: The Z-test is used, with the test statistic value given as \( z = -1.195 \).

### Part (a): Decision Table for Type I and Type II Errors
The table in part (a) requires us to classify the result of the test as correct, a Type I error, or a Type II error based on whether we reject or do not reject \( H_0 \) and the true nature of \( H_0 \).

1. **Type I Error**: Rejecting \( H_0 \) when \( H_0 \) is actually true.
2. **Type II Error**: Failing to reject \( H_0 \) when \( H_0 \) is actually false.
3. **Correct Decision**: Either rejecting \( H_0 \) when it’s false or not rejecting \( H_0 \) when it’s true.

Using this information, you can fill out the table as follows:

- **Reject \( H_0 \) when \( H_0 \) is true**: Type I Error
- **Reject \( H_0 \) when \( H_0 \) is false**: Correct Decision
- **Do not reject \( H_0 \) when \( H_0 \) is true**: Correct Decision
- **Do not reject \( H_0 \) when \( H_0 \) is false**: Type II Error

### Part (b): Z-Test P-Value Calculation
The Z-test statistic is \( z = -1.195 \). For a two-tailed test:

1. **Find the p-value**: Using a Z-table, find the p-value associated with \( z = -1.195 \) and double it because this is a two-tailed test.
   - For \( z = -1.195 \), the one-tailed p-value is approximately 0.116.
   - For a two-tailed test, the p-value is \( 2 \times 0.116 = 0.232 \).

2. **Decision based on p-value**: Since \( 0.232 > 0.10 \), we **do not reject \( H_0 \)** at the 10% significance level.

You can shade the central area under the normal curve (symmetrically around zero) up to \(-1.195\) and \(1.195\) to represent the p-value.

### Part (c): Conclusion for Hypothesis Test
Based on the p-value (0.232), which is greater than the significance level (0.10), the result is:
- **Do not reject the null hypothesis**.

### Part (d): Interpreting the Result (Type I or Type II Error)
If the true proportion is 50%, then \( H_0 \) is true. By not rejecting \( H_0 \), the decision is correct, not an error.

The answer to (d) is:
- **The result is correct**.

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### Do you need more details or have any questions?

#### Follow-Up Questions
1. Can you explain why we use a two-tailed test instead of a one-tailed test here?
2. How would the result change if the significance level were set to 5% instead of 10%?
3. What is the difference between a Type I and a Type II error in this context?
4. How is the p-value calculated from the z-score in hypothesis testing?
5. What would be the interpretation if the p-value were less than the significance level?

#### Tip:
Remember, in hypothesis testing, always check if the test is one-tailed or two-tailed, as this impacts how you calculate and interpret the p-value.

A sports league performs a hypothesis test at the 0.10 level of significance to determine if the proportion, p, of all times a certain referee’s coin comes up heads is different from 50%. The null hypothesis (H0) is p = 0.5, and the alternative hypothesis (H1) is p ≠ 0.5. The problem requires completing a Type I and Type II error table, performing a Z-test, determining whether to reject or not reject the null hypothesis, and identifying if the conclusion is correct if the true proportion is 50%.

Learn how to perform a hypothesis test at a 10% significance level to determine if a referee’s coin toss results in heads 50% of the time. This example covers null and alternative hypotheses, Type I and Type II errors, p-value calculations, and the interpretation of Z-test results. Suitable for college-level statistics.

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