Let’s address the problem step by step:

Question Analysis:

We are performing a hypothesis test for a population proportion.

Given Data:

• Sample size ($n$) = 745
• Number of successes ($x$) = 589
• Sample proportion ($\hat{p} = \frac{x}{n} = \frac{589}{745}$) ≈ 0.7906
• Claimed population proportion ($p_0$) = 0.75
• Significance level ($\alpha$) = 0.01

Part a) What is the null hypothesis?

$H_0: p = 0.75$ The null hypothesis states that the population proportion is 75%.

Part b) What is the alternative hypothesis?

$H_a: p \neq 0.75$ The alternative hypothesis is that the population proportion is not equal to 75%.

Part c) Is the test two-tailed, left-tailed, or right-tailed?

This is a two-tailed test because the alternative hypothesis ($H_a: p \neq 0.75$) suggests a deviation in either direction from 0.75.

Part d) Find the test statistic.

The test statistic for a proportion is calculated using the formula: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

Substitute the values: $z = \frac{0.7906 - 0.75}{\sqrt{\frac{0.75 \times (1 - 0.75)}{745}}}$ $z = \frac{0.0406}{\sqrt{\frac{0.75 \times 0.25}{745}}}$ $z = \frac{0.0406}{\sqrt{0.0002517}} = \frac{0.0406}{0.01587} \approx 2.56$

The test statistic is approximately $z = 2.56$.

Part e) Find the p-value.

For a two-tailed test, the p-value is calculated as: $p = 2 \times P(Z > |z|)$

Using $z = 2.56$ and a standard normal distribution table or calculator: $P(Z > 2.56) = 0.0052$ $p = 2 \times 0.0052 = 0.0104$

The p-value is approximately $p = 0.0104$.

Part f) Write an initial conclusion for the hypothesis.

Compare the p-value ($0.0104$) to the significance level ($\alpha = 0.01$):

• $p > \alpha$: Fail to reject $H_0$.

The initial conclusion is that there is insufficient evidence to reject the null hypothesis at the 0.01 significance level.

Part g) Write a final conclusion for the hypothesis test.

Since we failed to reject $H_0$, the data does not provide strong evidence that the true population proportion differs from 75%. Therefore, we conclude that it is plausible that 75% of adults believe it is morally wrong not to report all income on tax returns.

Questions for Further Exploration:

1. What happens if the significance level is changed to 0.05? Would the conclusion differ?
2. How does the sample size impact the reliability of this test?
3. Why is it important to use a two-tailed test in this scenario?
4. What would be the test statistic if the sample size were doubled but the proportions remained the same?
5. How would a higher claimed population proportion ($p_0$) affect the outcome?

Tip:

Always ensure you verify the assumptions for a hypothesis test, such as the sample size being large enough for the normal approximation to be valid.