Math Problem Statement

According to the 2010​ Census,

11.411.4​%

of all housing units in the United States were vacant. A county supervisor wonders if her county is different from this. She randomly selects

814814

housing units in her county and finds that

126126

of the housing units are vacant. Complete parts a through d below.

Question content area bottom

Part 1

​a) State the hypotheses.

Let the proportion of housing units in the county that are vacant be denoted by p. Choose the correct answer below.

A.Upper H 0 : p greater than or equals 0.114 Upper H Subscript Upper A Baseline : p less than 0.114

Upper H 0 : p greater than or equals 0.114H0: p≥0.114

Upper H Subscript Upper A Baseline : p less than 0.114HA: p<0.114

B.Upper H 0 : p equals 0.114 Upper H Subscript Upper A Baseline : p less than 0.114

Upper H 0 : p equals 0.114H0: p=0.114

Upper H Subscript Upper A Baseline : p less than 0.114HA: p<0.114

C.Upper H 0 : p less than 0.114 Upper H Subscript Upper A Baseline : p equals 0.114

Upper H 0 : p less than 0.114H0: p<0.114

Upper H Subscript Upper A Baseline : p equals 0.114HA: p=0.114

D.Upper H 0 : p equals 0.114 Upper H Subscript Upper A Baseline : p not equals 0.114

Upper H 0 : p equals 0.114H0: p=0.114

Upper H Subscript Upper A Baseline : p not equals 0.114

Solution

To test if the proportion of vacant housing units in the county differs from the national rate of 11.4% (or 0.114), the appropriate hypotheses are:

  • Null Hypothesis (H0): p=0.114p = 0.114
  • Alternative Hypothesis (HA): p≠0.114p \neq 0.114

This setup checks if the county’s vacancy rate is different (either higher or lower) from the national average. The correct answer is:

Answer: D.

H0:p=0.114andHA:p≠0.114H_0: p = 0.114 \quad \text{and} \quad H_A: p \neq 0.114

Would you like help with the next parts of the problem?

Relative Questions

  1. How is a hypothesis test for proportions different from one for means?
  2. What does it mean if we reject the null hypothesis in this context?
  3. Why is a two-tailed test appropriate here instead of a one-tailed test?
  4. How do we calculate the sample proportion of vacant units for the county?
  5. What level of significance is commonly used in hypothesis testing?

Tip: Always make sure to clearly define hypotheses to set up the test correctly.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Proportion Testing

Formulas

Test statistic for proportions: z = (p̂ - p0) / √(p0(1 - p0) / n)

Theorems

Central Limit Theorem
Hypothesis Testing for Proportions

Suitable Grade Level

College-level Statistics

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