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.

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.