Overview
This lecture demonstrates how to perform a one-proportion Z-test to determine if West Valley College's math success rate is significantly higher than the statewide rate, using both manual calculation and a graphing calculator.
Hypotheses Setup
- Null hypothesis (H₀): P = 0.54 (statewide math success rate in Fall 2010)
- Alternative hypothesis (H₁): P > 0.54 (West Valley's rate is significantly higher)
Sample Data & Proportion
- Sample size (n): 200 West Valley math students
- Number of successes (x): 133 students successful in math class
- Sample proportion (p̂): x/n = 133/200 = 0.665
Z-Test Statistic Calculation
- Formula: Z = (p̂ − P₀) / sqrt[P₀(1 − P₀) / n]
- Plug in values: Z = (0.665 − 0.54) / sqrt[0.54 × 0.46 / 200]
- Purpose: Compare sample proportion to hypothesized population proportion, adjusting for sample size
Calculator Shortcut (One-Proportion Z-Test)
- Use graphing calculator: STAT → TESTS → 1-PropZTest
- Input required: P₀ = 0.54, x = 133, n = 200, alternative: >
- Output: Z-test statistic (Z = 3.55), p-value (e.g., 0.000194 or 1.94e-4)
- Always choose the alternative hypothesis direction (greater than)
Key Terms & Definitions
- Null Hypothesis (H₀) — The statement that there is no effect or difference (P = 0.54)
- Alternative Hypothesis (H₁) — The statement we're testing for, i.e., P > 0.54
- Sample Proportion (p̂) — Proportion of successes in the sample (x/n)
- Z-Test Statistic — Value measuring how many standard errors the sample proportion is from the null
- P-value — Probability of observing the test statistic or something more extreme, given the null is true
Action Items / Next Steps
- Practice entering data into the calculator for one-proportion Z-tests
- Write down the p-value for later discussion
- Review process for identifying hypotheses and appropriate calculator function