Sampling Distribution of Proportions

Statistics · Axiom Academy

LESSON Sampling Distribution of Proportions Understanding the distribution of sample proportions and when to use normal approximation The sample proportion (denoted as p̂, pronounced "p-hat") is the proportion of successes observed in a random sample. If we survey 100 voters and 62 support a particular candidate: 2 Mean of the Sampling Distribution If we took many random samples and calculated p̂ for each, the mean of all those sample proportions would equal the true population proportion. The sample proportion p̂ is an unbiased estimator of the population proportion p. On average, it hits the target! The standard error (SE) measures how much sample proportions vary from sample to sample. Larger n → Smaller SE (more precise estimates) p closer to 0.5 → Larger SE (maximum variability) p closer to 0 or 1 → Smaller SE (less variability) 4 Normal Approximation Conditions When certain conditions are met, the sampling distribution of p̂ is approximately normal, allowing us to use z-scores and normal probability calculations. The sampling distribution of p̂ is approximately normal when: np ≥ 10 : Expected number of successes n(1-p) ≥ 10 : Expected number of failures Both conditions must be satisfied! 5 Summary: Sampling Distribution of p̂ When to Use Normal Approximation: Scenario: 40% of voters support a measure (p = 0.40). We sample n = 100 voters. n(1-p) = 100(0.60) = 60 ≥ 10 ✓ p̂ ~ N(0.40, 0.049) approximately

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