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This image presents a statistical problem related to hypothesis testing for comparing proportions from two different cities on the issue of raising the minimum wage. I'll walk you through the solution steps.

### Step 1: Identify the type of hypothesis test.
Since we are comparing the proportions of two independent samples (the voters from two different cities), we will conduct a **two-proportion z-test**.

### Step 2: Calculate the sample proportions.
- For **City 1**:
  \[
  \hat{p}_1 = \frac{350}{800} = 0.4375
  \]
- For **City 2**:
  \[
  \hat{p}_2 = \frac{384}{1000} = 0.384
  \]

### Step 3: Calculate the pooled sample proportion.
The pooled proportion \( \hat{p} \) combines the results from both samples:
\[
\hat{p} = \frac{350 + 384}{800 + 1000} = \frac{734}{1800} \approx 0.4078
\]

### Step 4: Compute the test statistic.
The formula for the z-statistic in a two-proportion z-test is:
\[
z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}
\]
Where:
- \( n_1 = 800 \)
- \( n_2 = 1000 \)
- \( \hat{p}_1 = 0.4375 \)
- \( \hat{p}_2 = 0.384 \)
- \( \hat{p} = 0.4078 \)

Now, let’s calculate the z-statistic.

The calculated z-statistic is approximately **2.30**.

### Step 5: Calculate the p-value.
To determine the p-value, we look at the standard normal distribution for the calculated z-statistic. Since this is a two-tailed test, we need to find the area in both tails.

Let me calculate the p-value.The p-value is approximately **0.0217**.

### Step 6: Compare the p-value to the significance level (α = 0.01).
Since the p-value (0.0217) is greater than the significance level (α = 0.01), **we do not have sufficient evidence** to reject the null hypothesis. Therefore, we cannot conclude that the level of support for raising the minimum wage differs significantly between the two cities at the 0.01 significance level.

### Summary:
- **Hypothesis test:** Two-proportion z-test.
- **Proportion in City 1 (p1):** 0.4375
- **Proportion in City 2 (p2):** 0.384
- **Pooled proportion (p):** 0.4078
- **Test statistic (z):** 2.30
- **p-value:** 0.0217
- **Conclusion:** No sufficient evidence to support a difference at α = 0.01.

Would you like more details on any step or further clarifications?

---

Here are 5 related questions:
1. How does changing the sample size affect the p-value in a two-proportion z-test?
2. What would happen if we used a significance level of 0.05 instead of 0.01?
3. How would the results change if City 1 had 400 supporters instead of 350?
4. What is the difference between a one-tailed and a two-tailed hypothesis test?
5. Can we apply this test if the sample sizes were significantly smaller?

**Tip:** Always define your null and alternative hypotheses clearly before starting the test to avoid errors in interpretation.

A political campaign is interested in whether a geographic difference existed in support for raising the minimum wage in a certain state. Polls were conducted in the two largest cities in the state about raising the minimum wage. In city 1, a poll of 800 randomly selected voters found that 350 supported raising the minimum wage. In city 2, a poll of 1000 randomly selected voters found that 384 supported raising the minimum wage. Does sufficient evidence exist to support the claim that the level of support differs between the two cities at the 0.01 significance level?

This problem involves conducting a two-proportion z-test to determine if there is a significant difference in support for raising the minimum wage between two cities. City 1 has a sample proportion of 0.4375, while City 2 has a sample proportion of 0.384. Using a significance level of 0.01, the problem explores whether the observed difference in proportions is statistically significant.

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