Testing a Claim About a Proportion

Overview

This topic explains how to test a claim about a population proportion using hypothesis testing. To perform a hypothesis test about a proportion, several conditions must be met:

• The sample must be a simple random sample.

• The conditions for a binomial distribution must be satisfied:

  • Fixed number of trials

  • Independent outcomes

  • Each outcome is classified as success or failure

  • Constant probability of success for each trial

• The normality conditions must be met: both and , where .

Key Concepts

• When testing a claim about a population proportion, two main methods are used:

  • p-Value Method: Compares the area (probability) under the curve for the test statistic to the significance level.

  • Critical Value Method: Compares a standardized test statistic to a critical region.

• Both methods lead to the same conclusion.

• The significance level () is the maximum probability of making a Type I error (rejecting a true null hypothesis).

Test Statistic Formula

The test statistic for a proportion is calculated as:

• = sample proportion =

• = claimed population proportion (from the null hypothesis)

• = sample size

Calculator Functions (TI-83/84)

For hypothesis testing involving proportions, the following calculator functions are used:

• "normalcdf": Converts a z-score to an area (for p-value method).

• "invNorm": Converts an area to a z-score (for critical value method).

• "1-PropZTest": Shortcut method to calculate the test statistic and p-value directly.

Example 1 – OxyContin Study

Scenario: OxyContin is tested for pain but can cause nausea. In a clinical trial, 227 subjects were surveyed, and 32 developed nausea. Test at a 0.05 significance level whether more than 20% of OxyContin users develop nausea.

1. State the Claim: The claim: More than 20% develop nausea.

2. Opposite of the Claim:

3. Define Hypotheses: Null hypothesis (): ; Alternative hypothesis (): (right-tailed test)

4. Significance Level:

5. Identify Test Statistic: Use the formula above.

6. Check Conditions: ; (both > 5)

7. Compute the Test Statistic:

8. Calculate z:

9. Determine the Critical Value: (right-tailed test),

10. Compare Results: Since , fail to reject .

11. p-Value Method: -value = P() = 0.981; , fail to reject .

12. Conclusion: There is insufficient evidence to support the claim that more than 20% of OxyContin users experience nausea.

Example 2 – Stem Cell Survey

Scenario: Adults were randomly selected for a Newsweek poll about stem cell funding. 481 were in favor, 401 against, and 120 unsure. A politician claimed that responses were random (like a coin toss). Exclude unsure participants and test at whether the proportion in favor equals 0.5.

1. State the Claim: Proportion in favor ;

2. Opposite of the Claim:

3. Test Statistic: ;

4. Calculate z:

5. p-Value: -value = 0.0076

6. Decision Rule: Since , reject .

7. Conclusion: There is sufficient evidence to reject the claim that responses are random (like flipping a coin). Therefore, the politician's claim is not supported.

Summary of Steps for Testing a Claim About a Proportion

1. State the claim (in words and symbols).

2. State the null and alternative hypotheses.

3. Identify the test statistic and significance level.

4. Check the normality conditions (, ).

5. Compute the test statistic ().

6. Use either the p-value method or the critical value method.

7. Compare the test statistic or p-value to or .

8. State the conclusion in nontechnical terms.

Key Formulas and Calculator Tips

• Test Statistic:

• Calculator Functions (TI-83/84):

  • "normalcdf": z-score to area (for p-value method)

  • "invNorm": area to z-score (for critical value method)

  • "1-PropZTest": shortcut for test statistic and p-value