Overview
This lecture covers hypothesis testing for a population proportion, including how to set up hypotheses, check assumptions, calculate test statistics, critical values, and interpret the p-value.
Setting Up Hypotheses
- The claim is that the population proportion (π) is less than 10% (π < 0.1).
- π represents the population proportion; some books use P.
- The alternative hypothesis (Hₐ) is π < 0.1; the null hypothesis (H₀) is π = 0.1.
- A "less than" alternative means the test is left-tailed.
- The direction of Hₐ always determines the test type, not the null hypothesis.
Checking Assumptions
- Assess if the sample is random or at least representative; in this example, the data was a census of a semester, assumed representative.
- Individuals in the sample should ideally be independent; this assumption may be questionable here.
- There must be at least 10 successes (cases with the characteristic) and 10 failures (cases without); here, 30 successes and 302 failures meet the requirement.
Calculating Test Statistic
- The test statistic formula is: (P̂ - π) / standard error, where P̂ is sample proportion.
- Standard error for a hypothesis test uses population proportion: √[π(1-π) / n].
- Plugging in values: (0.09036 - 0.1) / √[0.1 × 0.9 / 332] = -0.585.
- A negative z-score indicates P̂ is lower than π.
Interpreting Statistical Significance
- Critical value for a 5% significance level in a left-tailed test is -1.645.
- The test statistic (-0.585) is not beyond the critical value, so the result is not significant.
- The difference between sample and null proportion is not large enough to reject the null.
Calculating and Interpreting the p-value
- The p-value is the area in the tail left of the test statistic; calculated here as 0.279 (27.9%).
- A p-value much larger than 0.05 (significance level) means we do not reject the null hypothesis.
- High p-value suggests differences could be due to random chance (sampling variability).
Key Terms & Definitions
- Population Proportion (π or P) — The percentage of the entire population with a certain characteristic.
- Null Hypothesis (H₀) — The default assumption, usually states no difference or effect; uses the "=" symbol.
- Alternative Hypothesis (Hₐ) — The claim being tested, typically involves "<", ">", or "≠".
- Left-Tailed Test — Hypothesis test where the alternative hypothesis uses "<".
- Standard Error — Estimate of the standard deviation of the sample statistic, calculated from the null proportion.
- Test Statistic — Standardized value comparing sample statistic to null hypothesis, measured in standard errors.
- Critical Value — Threshold value; if the test statistic crosses it, the result is significant.
- p-value — Probability, assuming H₀ is true, of observing a result as extreme as or more extreme than the sample.
Action Items / Next Steps
- Practice setting up hypotheses and checking assumptions for one-proportion z-tests.
- Review critical values and calculating z-scores for left-, right-, and two-tailed tests.
- Complete any assigned problems on hypothesis testing for proportions.