AP Statistics 3.1-3.2: Estimators and Sampling Distributions

What is an estimator?

An estimator is a statistic used to estimate a population parameter. For example, the sample proportion $\hat p$ estimates the population proportion $p$.

What is the difference between an estimator and an estimate?

An estimator is the random statistic used to estimate a parameter, while an estimate is the numerical value obtained from one particular sample.

What does it mean for an estimator to be unbiased?

An estimator is unbiased if its sampling distribution is centered at the parameter it estimates. For an estimator $T$ of a parameter $\theta$, unbiasedness means $E(T)=\theta$.

What is bias in an estimator?

Bias is the difference between the mean of an estimator's sampling distribution and the parameter it estimates: $\operatorname{Bias}(T)=E(T)-\theta$. An unbiased estimator has bias $0$.

What is sampling variability?

Sampling variability is the natural sample-to-sample variation in a statistic when random samples of the same size are taken from the same population.

What is a sampling distribution?

The sampling distribution is the probability distribution of a statistic calculated from all possible random samples of a fixed size drawn from the same population. It describes how the statistic varies from sample to sample.

What four factors determine the sampling distribution of a statistic?

It depends on the population distribution, the statistic being calculated, the sampling method, and the sample size.

How does a sampling distribution differ from the distribution of individual population observations?

A population distribution describes individual observations, whereas a sampling distribution describes values of a statistic, such as a sample proportion, across many possible samples.

Why are sampling distributions useful for statistical inference?

They allow conclusions about a population to be based on the probability behavior of a statistic rather than on the complicated joint behavior of every individual observation.

What is the standard error of a statistic?

The standard error is the standard deviation of the statistic's sampling distribution. It measures the typical sampling-to-sampling variation in the statistic.

What is the sampling distribution of the number of successes in $n$ independent Bernoulli trials with success probability $p$?

The number of successes $X$ follows $X\sim\operatorname{Binomial}(n,p)$. The sample proportion is $\hat p=X/n$, so $n\hat p\sim\operatorname{Binomial}(n,p)$.

What are the mean and standard error of a sample proportion from independent Bernoulli trials?

For $\hat p=X/n$, the mean is $E(\hat p)=p$, so $\hat p$ is an unbiased estimator of $p$. Its standard error is $SE_{\hat p}=\sqrt{p(1-p)/n}$.

How does increasing the sample size affect the sampling distribution of $\hat p$?

The center remains at $p$, but the spread decreases because $SE_{\hat p}=\sqrt{p(1-p)/n}$. Multiplying $n$ by 4 cuts the standard error in half.

What conditions support using a normal approximation for the sampling distribution of $\hat p$?

The sample should be randomly selected, observations should be approximately independent, and the expected numbers of successes and failures should each be at least 10: $np\ge 10$ and $n(1-p)\ge 10$.

Why is the 10% condition used when sampling without replacement from a finite population?

The sample size should be no more than 10% of the population size, $n\le 0.10N$. This makes observations approximately independent because selecting one individual has little effect on the probabilities for the remaining selections.

Why is a random sample or random assignment important when studying the sampling distribution of $\hat p$?

Random sampling gives each sample a chance to be selected and makes the sampling model representative of the population. Without randomness, bias or dependence can invalidate the sampling distribution.

What does the central limit theorem imply for the sampling distribution of a sample proportion?

For a sufficiently large random sample with enough expected successes and failures, the sampling distribution of $\hat p$ is approximately normal with mean $p$ and standard deviation $\sqrt{p(1-p)/n}$, even though $\hat p$ itself is based on discrete outcomes.

How can a sampling distribution be studied when only one sample is actually observed?

It can be derived theoretically from a probability model for the population and sampling process, or approximated using simulation or resampling methods.

How does a sampling distribution support uncertainty quantification?

Its center indicates the statistic's typical value, while its spread indicates sampling uncertainty. These features are used to construct standard errors, confidence intervals, and hypothesis tests.