Math Problem Statement

The mayor of a town has proposed a plan for the annexation of a new bridge. A political study took a sample of 15001500 voters in the town and found that 74%74% of the residents favored annexation. Using the data, a political strategist wants to test the claim that the percentage of residents who favor annexation is above 71%71%. State the null and alternative hypotheses.

Solution

To test the claim about the percentage of residents who favor annexation, we need to set up the null hypothesis (H0H_0) and the alternative hypothesis (HaH_a).

Hypotheses:

  1. Null Hypothesis (H0H_0): The percentage of residents who favor annexation is equal to 71%.
    H0:p=0.71H_0: p = 0.71

  2. Alternative Hypothesis (HaH_a): The percentage of residents who favor annexation is greater than 71%.
    Ha:p>0.71H_a: p > 0.71

Explanation:

  • The null hypothesis represents the status quo or no effect—in this case, that the proportion of residents favoring annexation is 71%.
  • The alternative hypothesis represents the claim or research question we are testing—that the proportion is greater than 71%.
  • This is a one-tailed test because the strategist is specifically testing for an increase (greater than 71%).

Do you want to proceed with calculating the test statistic and p-value? Let me know!


Here are 5 related questions to expand your understanding:

  1. What is the significance level (α\alpha) for this test?
  2. How is the test statistic calculated for a proportion hypothesis test?
  3. What is the critical value for a one-tailed test at a common significance level (e.g., 0.05)?
  4. How does sample size affect the hypothesis test in this scenario?
  5. What would the conclusion be if the p-value is less than α\alpha?

Tip: Always ensure your alternative hypothesis aligns with the

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Proportions
Statistics

Formulas

Null Hypothesis: H0: p = p0
Alternative Hypothesis (One-tailed test): Ha: p > p0
Z-test for proportion: z = (p̂ - p0) / √(p0(1 - p0) / n)

Theorems

Central Limit Theorem
Standard Normal Distribution

Suitable Grade Level

Grades 10-12, College Level