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.