AP Statistics:: Sampling Distributions

A SAMPLING DISTRIBUTION describes:

- **A)** The distribution of values taken by a statistic (like a sample mean or sample proportion) over all possible samples of a given size from the same population
- **B)** A distribution that never varies from sample to sample
- **C)** The distribution of the entire population
- **D)** The distribution of a single sample's raw data values

Why do different random samples of the same size, taken from the same population, typically produce DIFFERENT values of a sample statistic (like the sample mean)?

- **A)** Because the population itself changes between samples
- **B)** Because of natural sampling variability -- each sample consists of a different, randomly chosen subset of individuals, so the computed statistic will naturally vary somewhat from sample to sample even though the underlying population doesn't change
- **C)** Because random sampling always produces a biased result
- **D)** Because sample statistics are always identical across samples of the same size

A point estimate (statistic) is called UNBIASED if:

- **A)** It has zero variability across samples
- **B)** It is calculated using a non-random sampling method
- **C)** The mean of its sampling distribution equals the true population parameter it is estimating
- **D)** It always exactly equals the population parameter for any single sample

Is the sample mean \(\bar{x}\) an unbiased estimator of the population mean \(\mu\)?

- **A)** No, the sample mean is always biased
- **B)** Yes -- the sampling distribution of \(\bar{x}\) is centered exactly at \(\mu\), regardless of sample size
- **C)** Only when the population is Normally distributed
- **D)** Only for very large sample sizes

Is the sample proportion \(\hat{p}\) an unbiased estimator of the population proportion \(p\)?

- **A)** No, \(\hat{p}\) always underestimates \(p\)
- **B)** Unbiasedness only applies to means, never to proportions
- **C)** Yes, the sampling distribution of \(\hat{p}\) is centered at \(p\)
- **D)** No, \(\hat{p}\) always overestimates \(p\)

The MEAN of the sampling distribution of the sample proportion \(\hat{p}\) equals:

- **A)** \(0.5\) always
- **B)** \(n\), the sample size
- **C)** \(p\), the true population proportion
- **D)** The sample proportion of a single specific sample

The STANDARD DEVIATION of the sampling distribution of \(\hat{p}\) is given by:

- **A)** \(\sigma_{\hat{p}} = np(1-p)\)
- **B)** \(\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}\)
- **C)** \(\sigma_{\hat{p}} = \dfrac{p}{n}\)
- **D)** \(\sigma_{\hat{p}} = \dfrac{p(1-p)}{n}\)

As the SAMPLE SIZE increases, what happens to the standard deviation of the sampling distribution of \(\hat{p}\) (assuming the population proportion \(p\) stays fixed)?

- **A)** It becomes undefined for large samples
- **B)** It stays exactly the same regardless of sample size
- **C)** It DECREASES, since larger samples produce more precise (less variable) estimates of the population proportion
- **D)** It increases