Sampling Distribution of a Sample Proportion

Sample Proportion Statistic

  • Is a random variable

  • Is represented by p̂ (“p-hat”)

  • Estimates population proportion parameter p

  • Has a distribution with mean and variance

  • Is related to the total number of successes (x).

  • Is an unbiased estimator of the population parameter p.

    • In the context of sampling distributions, "unbiased" refers to a sampling distribution that accurately represents the population parameter it is estimating.

    • It has a mean that is equal to the true population parameter being estimated.

    • On average, the estimates obtained from the sampling distribution are not systematically overestimating or underestimating the true population parameter.

Formulas

Expected Value or Mean of p̂ » E(p̂ ) = p

Standard Deviation of sampling distribution of p̂ » In the above formulas:

  • p is the population proportion.

  • n is the sample size.

Checks

Independent Sample Condition Check

  • The standard deviation of the sample proportion is valid if the assumption of independence is upheld

  • “Is sample size n less than 10% of the population size N?”

  • n < 0.1N (if true, passes check and can be assumed independent)

  • Note: if population size is only “large” then determine what N must be at least. If reasonable for the situation, write that as a statement and pass check.

Normal Approximation Check

  • np ≥ 10

  • nq ≥ 10 or n(1-p) ≥ 10

  • Only passes check if both of the above checks are true.