Math Problem Statement

The mayor of a town has proposed a plan for the construction of a new bridge. A political study took a sample of 900900 voters in the town and found that 60%60% of the residents favored construction. Using the data, a political strategist wants to test the claim that the percentage of residents who favor construction is more than 56%56%. State the null and alternative hypotheses.

Solution

In this scenario, you are dealing with a hypothesis test for proportions. The goal is to test whether the percentage of residents who favor construction is more than 56%.

Null and Alternative Hypotheses:

  • Null Hypothesis (H0H_0): The percentage of residents who favor construction is equal to 56%. H0:p=0.56H_0: p = 0.56
  • Alternative Hypothesis (HaH_a): The percentage of residents who favor construction is greater than 56%. Ha:p>0.56H_a: p > 0.56

Here, pp represents the true proportion of residents in the town who favor the construction of the bridge.

The test will be a one-tailed test (right-tailed) because the alternative hypothesis is testing whether the proportion is greater than 56%.

Would you like help on the next steps for testing this hypothesis?


Here are five questions that expand on this concept:

  1. What is the test statistic used for hypothesis testing of proportions?
  2. How do you calculate the p-value in a one-tailed test for proportions?
  3. What is the importance of the significance level (α\alpha) in hypothesis testing?
  4. How would you calculate the confidence interval for the proportion of voters favoring the bridge?
  5. How would the result change if the sample size were smaller?

Tip: Always check if the sample size is large enough to assume normality when performing hypothesis tests on proportions (use np5np \geq 5 and n(1p)5n(1-p) \geq 5 criteria).

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Proportions
One-tailed Test

Formulas

Null Hypothesis: H₀: p = 0.56
Alternative Hypothesis: Hₐ: p > 0.56

Theorems

Central Limit Theorem
Law of Large Numbers

Suitable Grade Level

Grades 11-12