Chapter 7.2 – Sampling Distributions: Sample Proportions

Vocabulary flashcards covering key terms, formulas, and concepts from Section 7.2 on sampling distributions of sample proportions.

Sample Proportion (p̂)

Statistic found by dividing the count of successes (X) by the sample size (n): p̂ = X / n.

Population Proportion (p)

Parameter representing the true proportion of successes in the entire population.

Sampling Distribution of p̂

The distribution of sample proportions obtained from all possible SRSs of a fixed size n drawn from the population.

Mean of p̂ (μ_p̂)

Equal to the population proportion: μ_p̂ = p, showing that p̂ is an unbiased estimator of p.

Standard Deviation of p̂ (σ_p̂)

Given by σ_p̂ = √[p(1 − p) / n] when the 10% condition is met.

10% Condition

Requirement that the sample size n is at most 10% of the population size N (n ≤ 0.1N) before using σ_p̂.

Normal Condition for p̂

The sampling distribution of p̂ is approximately Normal when both np ≥ 10 and n(1 − p) ≥ 10.

Shape–Center–Spread (p̂)

Shape becomes more symmetric as n increases; center is p; spread (σ_p̂) decreases as n grows.

Unbiased Estimator

A statistic whose sampling distribution has mean equal to the parameter it estimates (e.g., p̂ for p).

SRS (Simple Random Sample)

A sample in which every set of n individuals has an equal chance to be selected from the population.

Binomial Random Variable (X)

Counts the number of successes in n independent trials with probability p of success on each trial.

Mean of X (μ_X)

For a binomial variable: μ_X = np.

Standard Deviation of X (σ_X)

For a binomial variable: σ_X = √[np(1 − p)].

Relationship Between X and p̂

p̂ is a rescaled version of X: p̂ = (1/n) · X, so properties of X translate to p̂.

Effect of Sample Size (n)

Larger n makes σ_p̂ smaller and the sampling distribution more Normal and less skewed.

Skewness and p

Sampling distribution of p̂ is more skewed when p is near 0 or 1; more symmetric when p is near 0.5.

Planning for College Example

Illustrates using the Normal model of p̂ to find P(0.73 < p̂ < 0.87) when p = 0.80 and n = 125.

Z-Score for p̂

z = (p̂ − p) / σ_p̂; used to convert sample proportions to the standard Normal scale.

Simulation

Using repeated random sampling (e.g., Reese’s Pieces, pennies) to empirically model the sampling distribution.

Central Limit Principle for Proportions

As n increases, the distribution of p̂ approaches Normal, mirroring the CLT for means.