8. Sampling Distributions and the Central Limit Theorem Free Online FlashCards

What is a sampling distribution?

A sampling distribution is the probability distribution of a statistic calculated from many possible random samples of the same size from the same population.

How does a sampling distribution differ from a population distribution?

A population distribution describes individual observations, whereas a sampling distribution describes a statistic calculated across many samples.

What is the mean of the sampling distribution of \bar{X}?

The expected value of the sample mean equals the population mean: E(\bar{X})=\mu. Therefore, the sample mean is an unbiased estimator of \mu.

What is the standard error of a sample mean?

The standard error of a sample mean is SE(\bar{X})=\sigma/\sqrt{n}, where \sigma is the population standard deviation and n is the sample size.

How is the standard error of a mean estimated when \sigma is unknown?

When \sigma is unknown, the estimated standard error is \widehat{SE}(\bar{x})=s/\sqrt{n}, using the sample standard deviation s.

What does the Central Limit Theorem state?

The Central Limit Theorem states that, for independent observations with finite mean and variance, the sampling distribution of \bar{X} becomes approximately normal as n becomes sufficiently large.

What does the Central Limit Theorem not guarantee?

The CLT does not make individual observations normally distributed or make a biased sample representative. It concerns the distribution of a statistic under an appropriate sampling model.

What are the mean and standard error of \hat{p}?

For a sample proportion, E(\hat{p})=p and SE(\hat{p})=\sqrt{p(1-p)/n}. Here p is the population proportion and n is the sample size.

What condition commonly supports a normal approximation for \hat{p}?

A common check for an approximately normal sampling distribution of \hat{p} is np\ge10 and n(1-p)\ge10.

How do random sampling and random assignment differ?

Random sampling selects units from a population and supports generalization. Random assignment allocates participants to treatments and supports conclusions about cause and effect.

What happens to standard error when sample size is quadrupled?

Increasing n by a factor of 4 divides the standard error by 2 because standard error decreases at the rate 1/\sqrt{n}.

When is a finite-population correction appropriate?

For sampling without replacement from a finite population, the correction is \sqrt{(N-n)/(N-1)}, giving SE(\bar{X})=(\sigma/\sqrt{n})\sqrt{(N-n)/(N-1)}.