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The image contains several questions and concepts, likely from a statistics assignment or exam. Here's a step-by-step solution to each problem in the image:

1. **Question 1:** \( H_0 \) and \( H_a \)
   - The null hypothesis (\( H_0 \)): \( \mu = 0.5 \)
   - The alternative hypothesis (\( H_a \)): \( \mu > 0.5 \)
   
   To test this hypothesis at a 5% significance level, we compare the p-value with 0.05. If the p-value is less than 0.05, we reject \( H_0 \).

2. **Question 2:** Checking for normality and critical values
   - The z-critical value for a one-tailed test at 5% significance is 1.645.
   
   If the test statistic (z-score) is greater than 1.645, we reject the null hypothesis.

3. **Question 3:** Confidence intervals and hypothesis testing
   - A 95% confidence interval means we are 95% confident that the true mean lies within this range.
   
   For hypothesis testing, if the hypothesized mean (0.5) lies outside this interval, we reject the null hypothesis.

4. **Question 4:** Significance level and p-value interpretation
   - The significance level (\( \alpha \)) is 0.05.
   
   If the p-value is less than 0.05, we reject \( H_0 \). If it is greater than 0.05, we fail to reject \( H_0 \).

5. **Question 5:** Sample size and its effect on the confidence interval
   - Increasing the sample size decreases the margin of error, resulting in a narrower confidence interval.
   
   A larger sample size provides a more precise estimate of the population mean.

6. **Question 6:** Type I and Type II errors
   - Type I error (false positive): Rejecting \( H_0 \) when it is actually true.
   
   - Type II error (false negative): Failing to reject \( H_0 \) when it is false.
   
   The probability of a Type I error is equal to the significance level (\( \alpha \)).

7. **Question 7:** Power of a test
   - The power of a test is the probability of correctly rejecting \( H_0 \) when \( H_a \) is true.
   
   Higher power means a lower probability of a Type II error.

8. **Question 8:** Effect of standard deviation on the hypothesis test
   - A larger standard deviation increases the standard error, making it harder to reject \( H_0 \).
   
   Decreasing the standard deviation leads to a more precise test, increasing the chances of detecting a true effect.

Would you like further details or explanations on any of these points? If so, please let me know which one.

### Relative Questions:

1. What is the purpose of setting a significance level in hypothesis testing?
2. How do you calculate the z-critical value for different significance levels?
3. What are the steps to construct a confidence interval for a population mean?
4. How does the p-value help in making decisions about the null hypothesis?
5. Why is sample size important in statistical analysis?
6. What are the consequences of making a Type I error in a real-world scenario?
7. How can the power of a statistical test be increased?
8. In what ways does the standard deviation of a dataset affect the outcomes of hypothesis testing?

### Tip:
Always check assumptions (like normality) before performing hypothesis tests or constructing confidence intervals to ensure valid results.

Explore statistics concepts such as hypothesis testing, normal distribution, confidence intervals, and types of errors with detailed solutions. Suitable for advanced college-level students and professionals.

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