Notes from Testing a Claim - Section 9.2

Chapter 9: Testing a Claim

Section 9.2: Tests About a Population Proportion

Overview
  • Focus on significance tests for a population proportion.
  • Learning Targets:
    • STATE and CHECK the conditions for performing a significance test about a population proportion.
    • CALCULATE the standardized test statistic and P-value for a test about a population proportion.
    • PERFORM a significance test about a population proportion.

Conditions for Performing a Significance Test About a Proportion

  1. Random Condition:

    • Data must come from a random sample from the population of interest.

  2. 10% Condition:

    • When sampling without replacement, the sample size (n) should be less than 10% of the population size (N), i.e., n<0.10Nn < 0.10N.

  3. Large Counts Condition:

    • Both np<em>0np<em>0 and n(1−p</em>0)n(1 - p</em>0) must be at least 10, where p0p_0 is the population proportion from the null hypothesis.
Example Scenario: High School Part-Time Jobs
  • Problem Statement:
    • According to the U.S. Census Bureau, the proportion of high school students with part-time jobs is 0.25. An administrator suspects this proportion is lower at her school.
    • Null Hypothesis: H0:p=0.25H_0: p = 0.25
    • Alternative Hypothesis: Ha:p<0.25H_a: p < 0.25
    • Significance level α=0.05α = 0.05.
    • Sample size: 200 students, with 39 having part-time jobs.
Checking Conditions
  • Random: ✓ Random sample of 200 students.
  • 10%: ✓ 200 is less than 10% of students at a large high school.
  • Large Counts:
    • np0=200imes0.25=50ext(sufficientsince50ext≥10)np_0 = 200 imes 0.25 = 50 ext{ (sufficient since } 50 ext{ } ≥ 10).
    • n(1−p0)=200imes(1−0.25)=150ext(sufficientsince150ext≥10)n(1 - p_0) = 200 imes (1 - 0.25) = 150 ext{ (sufficient since } 150 ext{ } ≥ 10).

Standardized Test Statistic

  • Standardized test statistic is given by:
    z=p^−p<em>0p</em>0(1−p0)nz = \frac{\hat{p} - p<em>0}{\sqrt{\frac{p</em>0(1 - p_0)}{n}}}
  • Where:
    • p^\hat{p} is the sample proportion.
Hypothetical Basketball Shooter Claim
  • Problem Statement:
    • A basketball player claims to be an 80% free-throw shooter: H<em>0:p=0.80H<em>0: p = 0.80; H</em>a:p<0.80H</em>a: p < 0.80.
  • Sample Expected Values:
    • For a sample of 50 shots, we expect the sample proportion around p^=0.80\hat{p} = 0.80.
Calculating Variability
  • Variability is calculated using:
    σ<em>p^=p</em>0(1−p0)n=0.80(1−0.80)50=0.0566\sigma<em>{\hat{p}} = \sqrt{\frac{p</em>0(1 - p_0)}{n}} = \sqrt{\frac{0.80(1 - 0.80)}{50}} = 0.0566
Example Calculations

  1. Sample Observation:

    • Player makes 32 out of 50 shots. Thus, p^=3250=0.64\hat{p} = \frac{32}{50} = 0.64.

  2. Standardized Test Statistic Calculation:

    • z=0.64−0.800.0566=−2.83z = \frac{0.64 - 0.80}{0.0566} = -2.83.
    • Descriptor: The value z=−2.83z = -2.83 measures how far the sample statistic is from what is expected under the null hypothesis in standard deviation units.
P-value Calculation
  • Use Table A or technological methods for calculation.
  • For z=−2.83z = -2.83, lookup or calculate P-value (area in the tail):
    P(z≤−2.83)=0.0023P(z ≤ -2.83) = 0.0023
Interpretation of Results
  • If P-value is less than αα, reject the null hypothesis. For instance, with a P-value of 0.0023 which is less than 0.05, there is significant evidence against the null hypothesis.
Putting It All Together: One-Sample Z-Test Steps
  1. State:
    • Clearly articulate the hypotheses and significance level.
  2. Plan:
    • Identify the appropriate inference method and check conditions.
  3. Do: If conditions are satisfactory, carry out the analysis:
    • State sample statistics, calculate standardized test statistics and find P-values.
  4. Conclude: Make informed conclusions related to hypothesis testing.
Conditions for Validity
  1. Random Condition
    • Ensures p^−p0\hat{p} - p_0 is a valid estimator of the population difference.
  2. Large Counts Condition
    • Enables Normal distribution usage for modeling p^\hat{p}.
  3. 10% Condition
    • Justifies using the typical formulas under sampling without replacement.
Problem Example: Quality Control in Potato Shipment
  • Context:
    • Testing proportion of potatoes with blemishes.
    • Null Hypothesis: H<em>0:p=0.08H<em>0: p = 0.08; Alternative: H</em>a:p>0.08H</em>a: p > 0.08.
    • Use a random sample of 500 potatoes with 47 blemishes observed.
Conclusion for Potato Shipment Test
  • Conclude based on the comparison of P-values to αα.
    • In this example, since P-value > 0.10, fail to reject the null.
Two-Sided Tests
  • In two-sided tests, use H<em>a:p≠p</em>0H<em>a: p ≠ p</em>0; find P-value as probability of extreme proportions in either direction.
Final Summary of Key Learning Targets
  • STATE and CHECK the Random, 10%, and Large Counts conditions.
  • CALCULATE the standardized test statistic and P-value.
  • PERFORM significance tests about a population proportion.