Module 19 Wrap-Up: Hypothesis Tests for a Population Proportion
Overview
This module summarizes the key steps and considerations for conducting a hypothesis test regarding claims about a population proportion.
Four Steps of a Hypothesis Test
Step 1: Determine the Hypotheses
- Null Hypothesis (H₀): Proposes that the population proportion ( p ) equals a specific value ( p_0 ).
- Alternative Hypothesis (Hₐ): Suggests the population proportion is either less than, greater than, or not equal to ( p_0 ).
Step 2: Collect the Data
- Essential to use random selection and random assignment if conducting an experiment.
- Verify the use of the normal curve to represent distribution:
- Both ( n \times p_0 ) and ( n \times (1 - p_0) ) should be at least 10.
Step 3: Assess the Evidence
- Calculate the test statistic (z-score):
- Formula: [ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} ]
- Determine the p-value using StatCrunch or similar tools:
- If Hₐ is greater than, p-value is the area to the right.
- If Hₐ is less than, p-value is the area to the left.
- If Hₐ is not equal, the p-value is double the tail area beyond the test statistic.
Step 4: Give the Conclusion
- Small p-value: Data unlikely under H₀, leading to rejection of H₀.
- P-value ≤ Significance Level: Reject H₀ and accept Hₐ.
- P-value > Significance Level: Fail to reject H₀.
- Conclusions should be in the context of the research question.
Errors in Hypothesis Testing
- Type 1 Error: Rejecting H₀ when it is true.
- Type 2 Error: Failing to reject H₀ when Hₐ is true.
- Errors occur due to random chance, not procedural mistakes.
Additional Considerations
- p-value Significance: Probability of the observed sample proportion if H₀ is true.
- Sample Size Impact:
- Larger samples increase the likelihood of rejecting H₀ if Hₐ is true.
- May detect insignificant differences if the sample size is too large.
- Data Quality:
- "Garbage in, garbage out": Poor collection methods render results meaningless.
- Results are specific to the sampled population.