Sampling and Estimation

Convenience sampling is a non-probability sampling method where subjects are selected based on their ease of access or convenience, rather than their representativeness of the population.

In a simple random sample, each individual in the population has an equal chance of being selected, so the probability of selecting any specific individual is 1/N, where N is the population size.

The sampling distribution of the mean is the probability distribution of all possible sample means that can be obtained from a population of a given size.

The sampling distribution of the sample mean follows a normal distribution with mean (\mu) and standard deviation (\sigma/\sqrt{n}), where n is the sample size.

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.

A point estimate is a single value that is used to estimate a population parameter. The sample mean is a point estimate of the population mean.

An interval estimate is a range of values that is used to estimate a population parameter. A confidence interval is an interval estimate of the population mean.

The width of a confidence interval decreases as the sample size increases, because a larger sample size provides more information about the population.

The level of confidence in a confidence interval is the probability that the interval contains the true population parameter.

A t-distribution is used to construct a confidence interval for a population mean when the sample size is small (less than 30) and the population standard deviation is unknown.

A z-distribution is used to construct a confidence interval for a population proportion when the sample size is large (greater than 30) and the population proportion is not close to 0 or 1.

A chi-square distribution is used to construct a confidence interval for a population variance when the sample size is large (greater than 30).

A t-distribution is used to construct a confidence interval for a population correlation coefficient when the sample size is small (less than 30).

A t-distribution is used to construct a confidence interval for a population regression coefficient when the sample size is small (less than 30).

A t-distribution is used to construct a confidence interval for a population mean difference when the sample sizes are small (less than 30) and the population standard deviations are unknown.