To determine the type of hypothesis test and perform the necessary calculations, let's go step by step.

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Type of Hypothesis Test

This is a two-tailed z-test because we are comparing the proportions of two independent groups (city 1 and city 2) to check if there is a significant difference between their support levels.

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Step-by-Step Calculations

1. Sample Proportions:

$p_1 = \frac{\text{Number of supporters in City 1}}{\text{Sample size of City 1}} = \frac{518}{800} = 0.6475$ $p_2 = \frac{\text{Number of supporters in City 2}}{\text{Sample size of City 2}} = \frac{611}{1000} = 0.611$

2. Combined Proportion ($p^*$):

$p^* = \frac{\text{Total supporters from both cities}}{\text{Total sample size from both cities}} = \frac{518 + 611}{800 + 1000} = \frac{1129}{1800} \approx 0.6272$

3. Standard Error ($SE$):

$SE = \sqrt{p^*(1 - p^*) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.6272(1 - 0.6272) \left( \frac{1}{800} + \frac{1}{1000} \right)}$ $SE \approx \sqrt{0.6272 \cdot 0.3728 \cdot (0.00125 + 0.001)} = \sqrt{0.6272 \cdot 0.3728 \cdot 0.00225} \approx \sqrt{0.000525} \approx 0.0229$

4. Test Statistic ($z$):

$z = \frac{p_1 - p_2}{SE} = \frac{0.6475 - 0.611}{0.0229} \approx \frac{0.0365}{0.0229} \approx 1.59$

5. p-value:

For a two-tailed test, we find the p-value corresponding to $z = 1.59$: $p\text{-value} = 2 \cdot P(Z > 1.59)$ Using a standard normal table or calculator: $P(Z > 1.59) \approx 0.0559$ $p\text{-value} = 2 \cdot 0.0559 = 0.1118$

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Conclusion:

At a significance level of $\alpha = 0.01$, the p-value ($0.1118$) is greater than $\alpha$.

Thus, we fail to reject the null hypothesis. There is insufficient evidence to support the claim that the level of support for raising the minimum wage differs between the two cities.

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Final Answers:

1. Type of hypothesis test: Two-tailed z-test
2. $p_1 = 0.6475$
3. $p_2 = 0.611$
4. $p^* = 0.6272$
5. Test statistic: $z \approx 1.59$
6. p-value: $0.1118$
7. Conclusion: No, sufficient evidence does not exist at $\alpha = 0.01$.

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Do you want further clarification or additional examples?
Here are 5 related questions to expand on this:

1. How would the test change if we used a one-tailed z-test?
2. What sample sizes are needed to reduce the p-value below 0.01?
3. How is the combined proportion $p^*$ used in hypothesis testing?
4. How does the significance level affect the decision in hypothesis testing?
5. What are the assumptions behind using a z-test for proportions?

Tip: Always check if the sample sizes are large enough for the z-test approximation to the normal distribution to be valid. A general rule is $np > 5$ and $n(1-p) > 5$.