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Let's break this down step by step to solve the problem from your exam.

### Question Details:
You are conducting a study to determine if the proportion of voters who prefer Candidate A differs from 56%, at a significance level of \( \alpha = 0.05 \). The sample includes 71 voters, and 40 out of those 71 prefer Candidate A.

We will go through each part of the question and find the appropriate answers.

### 1. Study Proportion \( p \):
For this study, you are testing if the true population proportion \( p \) is different from 0.56 (or 56%).

This is a two-tailed test since you want to check if it is "different," not specifically greater or less.

### 2. Null and Alternative Hypotheses:
- \( H_0: p = 0.56 \) (the population proportion is 56%)
- \( H_a: p \neq 0.56 \) (the population proportion is different from 56%)

### 3. Test Statistic:
The test statistic \( z \) is calculated using the formula for a proportion hypothesis test:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]

Where:
- \( \hat{p} \) = Sample proportion = \( \frac{40}{71} \approx 0.563 \)
- \( p_0 = 0.56 \) (null hypothesis proportion)
- \( n = 71 \) (sample size)

Now, let's compute this:

\[
z = \frac{0.563 - 0.56}{\sqrt{\frac{0.56(1 - 0.56)}{71}}} 
= \frac{0.003}{\sqrt{\frac{0.2464}{71}}}
= \frac{0.003}{\sqrt{0.00347}} 
= \frac{0.003}{0.059}
\]

So, \( z \approx 0.051 \).

### 4. P-value:
The p-value corresponds to the probability of observing a test statistic as extreme as the one calculated (in both directions, since it's two-tailed). Since the z-value is very small (near 0), the p-value will be close to 1.

To get an approximate p-value for \( z = 0.051 \), we can look this up in a standard normal distribution table or use a calculator. The p-value is about 0.959.

### 5. Compare P-value to \( \alpha \):
Since the p-value (0.959) is much greater than \( \alpha = 0.05 \), we **fail to reject** the null hypothesis.

### 6. Conclusion:
Based on the test, there is **not enough evidence** to suggest that the proportion of voters who prefer Candidate A is different from 56%.

Thus, the final answer should be:
- "The sample data suggest that the population proportion is not significantly different from 56% at \( \alpha = 0.05 \), so there is not sufficient evidence to conclude that the proportion of voters who prefer Candidate A is different from 56%."

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Do you want more details on any of the steps? Let me know if you have any questions!

### Related Questions:
1. How is a two-tailed test different from a one-tailed test?
2. What is the significance level, and how does it impact hypothesis testing?
3. How do we interpret the p-value in hypothesis testing?
4. Why do we use a z-test for proportions instead of other tests?
5. What is the relationship between the sample size and the test statistic?

### Tip:
Always double-check your p-values and z-scores when performing hypothesis tests, as they directly influence your conclusions.

You are conducting a study to see if the proportion of voters who prefer Candidate A is significantly different from 56% at a significance level of α = 0.05. According to your sample, 40 out of 71 potential voters prefer Candidate A.

Conduct a hypothesis test to determine if the proportion of voters supporting Candidate A is significantly different from 56%. The sample data includes 40 voters out of 71 who prefer Candidate A, and the test is performed at a 0.05 significance level using a Z-test for proportions.

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