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 $\tilde{p}$ ($\boldsymbol{\tau}_{\tilde{p}}$)

The center of the sampling distribution of the sample proportion; $\boldsymbol{\tau}_{\tilde{p}} = p$.

Standard deviation of $\boldsymbol{\tilde{p}}$ ($\tau_{\tilde{p}}$)

The spread of the sampling distribution of the sample proportion; $\tau_{\tilde{p}} = \sqrt\frac{p(1-p)}{n}$.

Standard error (SE)

An estimate of the standard deviation of a sampling distribution, often computed by plugging sample values (like $\tilde{p}$) 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 $\tilde{p}$

When conditions hold, $\tilde{p}$ is approximately Normal: $\tilde{p} \approx N(p, \sqrt\frac{p(1-p)}{n})$.

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 $n \le 0.1N$.

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 ($\bar{x}$)

The average of n numerical observations: \bar{x} = \frac{x_1 + x_2 + \boldsymbol{\text{···}} + x_n}{n}.

Population mean (μ)

The true average of a population; a parameter.

Mean of the sampling distribution of $\bar{x}$ ($\boldsymbol{\mu}_{\bar{x}}$)

The center of the sampling distribution of the sample mean; $\boldsymbol{\mu}_{\bar{x}} = \boldsymbol{\mu}$.

Standard deviation of $\bar{x}$ ($\boldsymbol{\tau}_{\bar{x}}$)

The spread of the sampling distribution of the sample mean; $\tau_{\bar{x}} = \frac{\tau}{\sqrt{n}}$.

Central Limit Theorem (CLT)

For large n, the sampling distribution of $\bar{x}$ becomes approximately Normal regardless of population shape (assuming a well-defined $\boldsymbol{\tau}$ and $\boldsymbol{\nu}$ 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); $\tilde{p}$ is the mean of indicator variables.

Sampling distribution of a sum ($S$)

For S = x_1 + \boldsymbol{\text{···}} + x_n: $\boldsymbol{\tau}_{S} = n\boldsymbol{\tau}$ and $\tau_{S} = \tau \sqrt{n}$ (under similar conditions as for $\bar{x}$).