Unit 5 Notes: Understanding Sampling Distributions (Proportions and Means)

Sampling distribution

The distribution of a statistic over all possible random samples of a fixed size from the same population.

Statistic

A number computed from a sample (e.g., sample mean or sample proportion) that varies from sample to sample.

Parameter

A fixed numerical value that describes a population (e.g., p or μ).

Sample proportion (p-hat, p̂)

The fraction of individuals in a sample with a particular characteristic (success); p̂ = X/n.

Population proportion (p)

The true proportion of the population that has a given characteristic; a parameter.

Success (in a proportion setting)

An outcome counted as having the characteristic of interest (coded as 1 in indicator-variable terms).

Unbiased estimator

A statistic whose sampling distribution is centered at the true parameter value (it does not systematically over- or underestimate).

Mean of the sampling distribution of p̂ (μ_p̂)

The center of the sampling distribution of the sample proportion; μ_p̂ = p.

Standard deviation of p̂ (σ_p̂)

The spread of the sampling distribution of the sample proportion; σ_p̂ = √(p(1−p)/n).

Standard error (SE)

An estimate of the standard deviation of a sampling distribution, often computed by plugging sample values (like p̂) into the SD formula.

Large Counts condition

Condition for Normal approximation of p̂: np ≥ 10 and n(1−p) ≥ 10 (using p when p is given).

Normal approximation for p̂

When conditions hold, p̂ is approximately Normal: p̂ ≈ N(p, √(p(1−p)/n)).

Random condition

Requirement that data come from a random sample or randomized experiment so probability-based results are trustworthy.

Independence condition

Requirement that individual outcomes are (approximately) independent, so sampling distribution formulas apply.

10% condition

When sampling without replacement from a finite population of size N, independence is reasonable if n ≤ 0.1N.

Sampling without replacement

Selecting items from a finite population with no repeats; can create dependence if the sample is a large fraction of the population.

z-score (standardization)

A standardized value measuring how many standard deviations a statistic is from its mean: z = (observed − mean)/SD.

Sample mean (x-bar, x̄)

The average of n numerical observations: x̄ = (x1 + x2 + ··· + xn)/n.

Population mean (μ)

The true average of a population; a parameter.

Mean of the sampling distribution of x̄ (μ_x̄)

The center of the sampling distribution of the sample mean; μ_x̄ = μ.

Standard deviation of x̄ (σ_x̄)

The spread of the sampling distribution of the sample mean; σ_x̄ = σ/√n.

Central Limit Theorem (CLT)

For large n, the sampling distribution of x̄ becomes approximately Normal regardless of population shape (assuming a well-defined μ and σ and approximate independence).

Normality condition for x̄

To use a Normal model for x̄: the population is Normal, or n is large enough for the CLT to give an approximate Normal sampling distribution.

Indicator variable

A 0–1 variable (1 = success, 0 = failure); p̂ is the mean of indicator variables.

Sampling distribution of a sum (S)

For S = x1 + ··· + xn: μS = nμ and σS = σ√n (under similar conditions as for x̄).