Let's analyze each part of the question from the uploaded images. The problem is a hypothesis test for two population proportions, with the goal of determining if a higher proportion of women watch local news compared to men.

Step 1: Define the Variables

• Since we are comparing proportions, the appropriate choice for the variable is: $p = \text{the proportion of women or men who watch local news.}$ Therefore, the correct option here is:
  • $p$ for women and $p$ for men.

Step 2: Formulate the Hypotheses

Given that we want to test if a higher proportion of women watch local news than men, the hypotheses should be structured as follows:

1. Null Hypothesis ($H_0$): $p_1 - p_2 = 0$

   • This implies there is no difference in the proportions of women and men who watch local news.

2. Alternative Hypothesis ($H_A$): $p_1 - p_2 > 0$

   • This implies that the proportion of women watching local news is greater than the proportion of men.

So, the selected hypotheses should be:

• $H_0: p_1 - p_2 = 0$
• $H_A: p_1 - p_2 > 0$

Step 3: Find the Test Statistic and P-Value

• Sample Proportions:

  • Proportion of women ($p_1$) = 61% = 0.61
  • Proportion of men ($p_2$) = 51% = 0.51

• Sample Sizes: $n_1 = 200$, $n_2 = 200$

• Difference in Sample Proportions: $p_1 - p_2 = 0.61 - 0.51 = 0.10$

To find the test statistic $z$: $z = \frac{(p_1 - p_2)}{\sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$ where: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$ Using this formula, you can calculate $z$ and then the corresponding p-value.

Step 4: Decision Rule

Using an alpha level of 5% (0.05):

• If the p-value is less than 0.05, reject $H_0$.
• Otherwise, fail to reject $H_0$.

Step 5: Conclusion

Based on the test result (p-value and z-score), you would conclude whether there is strong evidence that the proportion of women who watch local news is higher than that of men.

Would you like further assistance with the calculations, or do you need clarification on any part?

Related Questions

1. How do we calculate the pooled sample proportion $\hat{p}$ for two samples?
2. What assumptions must be met for a two-proportion z-test?
3. How does the choice of alpha level affect hypothesis testing results?
4. What is the interpretation of a p-value in hypothesis testing?
5. How do we interpret the outcome if $H_0$ is not rejected?

Tip

Always check assumptions, like normality and independent samples, before conducting a hypothesis test for proportions.