Sampling distributions practice questions (AP Statistics)

Sample questions

The one free AP Statistics question on this unit, from the free AP Statistics Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · Sampling distributions
   A population has standard deviation $\sigma$. If the sample size is increased from $n = 25$ to $n = 100$, the standard deviation of the sampling distribution of the sample mean:
   • A.
     is divided by 4
   • B.
     is divided by 2
   • C.
     is unchanged
   • D.
     is multiplied by 2
   • E.
     becomes zero
   Show answerFrom the free AP Statistics Practice Exam →

Key terms

The definitions and formulas AP Statistics expects for sampling distributions, typeset as they appear on the paper.

Parameter versus statistic

A parameter describes the population ($p$, $\mu$); a statistic describes a sample ($\hat{p}$, $\bar{x}$) and varies from sample to sample. Use the right symbol in every hypothesis.

Sampling distribution

The distribution of a statistic over all possible samples of a given size from the population. It is what makes probability statements about a single sample possible.

Sampling distribution of a sample proportion

Centered at $p$ with a standard deviation that shrinks with the square root of the sample size. Approximately Normal when the large counts condition holds, and the standard deviation formula needs the 10 percent condition.

$$\mu_{\hat{p}} = p, \qquad \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}, \qquad np \ge 10 \text{ and } n(1-p) \ge 10$$

Sampling distribution of a sample mean

Centered at $\mu$ with standard deviation $\frac{\sigma}{\sqrt{n}}$. Normal if the population is Normal, and approximately Normal for large samples by the Central Limit Theorem.

$$\mu_{\bar{x}} = \mu, \qquad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Central Limit Theorem

For a large enough sample (the course uses $n \ge 30$), the sampling distribution of the sample mean is approximately Normal whatever the shape of the population.

Unbiased estimator

A statistic whose sampling distribution is centered at the parameter it estimates. $\bar{x}$ and $\hat{p}$ are unbiased; variability, not bias, is what a larger sample reduces.

Frequently asked questions

Other AP Statistics topics

• Exploring one-variable data
• Exploring two-variable data
• Collecting data
• Probability, random variables and distributions
• Inference for proportions
• Inference for means
• Chi-square tests
• Inference for slopes

All AP Statistics topics · AP Statistics exam guide · All free practice exams