# Statistics Ch. 8 - Sampling Distributions

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Fundamentals of Statistics, 3rd ed - Sullivan

### sampling distribution

a probability distribution for all possible values of the statistic as computed from a sample of size n

### sampling distribution of the sample mean

the probability distribution of all possible values of the random variable x̄ computed from a sample size n from a population with mean µ and standard deviation σ

### standard error of the mean

the standard deviation of the sampling distribution of x̄; σ-sub-x̄ = σ/√n

### central limit theorem

regardless of the shape of the underlying population, the sampling distribution of x̄ becomes approximately normal as the sample size, n, increases

### sample proportion

p̂ = x/n where x is the number of individuals in the sample with the specified characteristic. The sample proportion, p̂, is a statistic that estimates the population proportion, p.

### p̂

sample proportion

### sampling distribution of p̂

For a simple random sample of size n with a population proportion p,
1. The shape of the sampling distribution of p̂ is approximately normal provided np(1 - p) ≥ 10
2. The mean of the sampling distribution of p̂ is µ-sub-p = p
3. The standard deviation of the sampling distribution of p̂ is
σ-sub-p = √[(p(1 - p))/n]

Example: