Sampling Distribution of the Sample Proportion Calculator

When taking repeated samples from a population, the proportion of successes in each sample (ŵ) varies predictably. According to the Central Limit Theorem, this sampling distribution approximates a normal distribution under certain conditions. The standard deviation of the sample proportion, often called the standard error, is calculated using the formula: σₗ = √[p(1 - p) / n], where p is the population proportion and n is the sample size. To find the probability of a specific sample proportion, raw proportions are converted into z-scores using z = (ŵ - p) / σₗ, which are then evaluated against standard normal distribution cumulative probabilities.

Imagine a political campaign manager who knows that the true population proportion (p) of voters supporting their candidate is 0.52 (52%). They want to know the probability that a random sample of n = 400 voters will yield a sample proportion greater than 55% (ŵ = 0.55). First, we calculate the standard deviation of the sample proportion: σₗ = √[0.52(0.48) / 400] = √[0.2496 / 400] = √0.000624 ≈ 0.02498. Next, we convert the target sample proportion into a z-score: z = (0.55 - 0.52) / 0.02498 = 0.03 / 0.02498 ≈ 1.20. Looking up a z-score of 1.20 in a standard normal distribution table gives a right-tail probability of approximately 0.1151. Therefore, there is an 11.51% chance that a random sample of 400 voters will show a support level of 55% or higher.

Always verify the success-failure condition before performing calculations: both np and n(1-p) should be greater than or equal to 10 to ensure the normal approximation is sufficiently accurate. Additionally, pay close attention to whether your query involves a strict inequality (greater than versus greater than or equal to), as continuous normal distributions yield identical values for both, but discrete continuity corrections might apply in specialized coursework.