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Sampling distribution of proportion; parameter vs statistic

Short answer

The sampling distribution of sample proportion p̂ is the distribution of p̂ = X/n from all samples of size n, with mean p and SE √(pq/n). A parameter is a fixed population measure (e.g., μ, p), while a statistic is a sample-based variable (e.g., x̄, p̂) used to estimate it.

What do you mean by the sampling distribution of sample proportion? Distinguish between parameter and statistics.

The sampling distribution of the sample proportion is the probability distribution of the values of p̂ = X/n obtained from all possible random samples of size n drawn from a population in which the proportion of "successes" is p.

For a population proportion p and q = 1 − p:

  • Mean of p̂: E(p̂) = p
  • Standard error: σ(p̂) = √(pq/n) (multiply by √((N−n)/(N−1)) for a finite population when n/N > 0.05)
  • Shape: if np ≥ 5 and nq ≥ 5 (large n), p̂ is approximately normal, so Z = (p̂ − p)/√(pq/n) ~ N(0, 1).

For example, if 30% of the components in a lot are defective (p = 0.3) and samples of n = 100 are taken, p̂ has mean 0.3 and standard error √(0.3×0.7/100) = 0.0458.

Parameter and statistic

BasisParameterStatistic
MeaningNumerical measure describing the populationNumerical measure computed from a sample
ValueFixed and usually unknownKnown, changes from sample to sample
NatureConstantRandom variable with its own sampling distribution
Symbolsμ, σ, σ², p, ρx̄, s, s², p̂, r
UseWhat we wish to knowUsed to estimate or test the parameter

Example: the mean life μ of all bulbs made by a factory is a parameter; the mean x̄ of 50 tested bulbs is a statistic used to estimate μ.

✓ Answer checked by Claude Sonnet 5.5

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