Sampling Distribution Lessons

When selecting a sample from a population, the method you use depends on whether you're sampling from a finite population or infinite population. If sampling from a finite population, you want to select what's called a simple random sample. A simple random sample is defined a sample selected such that every possible sample of the same size is equally likely. The way we can assure that each sample has the same likelihood of being selected is by using a random number generator. There are two different kinds of simple random samples. In sampling without replacement, the same element is not allowed to repeat in the sample. However, in sampling with replacement, the same element may appear more than once in the sample. Although sampling without replacement is the preferred method, sampling with replacement is a valid method of selecting a simple random sample. If sampling from an infinite population, we want to take a random sample rather than a simple random sample.

After a sample has been selected from the population, a procedure known as point estimation can be performed. Point estimation if the process of computing what's known as a sample statistic from the sample. Sample statistics are estimated of their corresponding population parameters. For example, the sample mean is a point estimate of the population mean, the sample standard deviatino is a point estimate of the population standard deviation, and so on. So the sample statistics are known as "point estimators" of their corresponding population parameters. The term "point" is used here because they provide single-value estimate, as opposed to interval estimate. The numerical values of the sample statistics are then known as "point estimates."

The probability distribution of a sample statistics is known as a sampling distribution. The idea is that the selection of a simple random sample is an experiment. This is because it is a process that generates some outcomes, specificaly the elements of the sample. The a sample statistic, such as the sample mean, provides a numerical description of the outcome of this experiment of sampling. So a sample statistic is a random variable. Like all random variables, the sample statistic has an expected value, standard deviation and probability distribution. The probability distribution of a sample statistic is known as a sampling distribution because it is the result of sampling.

The sampling distribution of $\bar{x}$ is the probability distribution of the sample mean. There are three things we need to know to fully describe the sampling distribution of the sample mean. These are the expected value, standarding deviation and the form of the sampling distribution. First, the expected value, or mean, of the sample mean is equal to the population mean. The standard deviation of the sample mean, known as the standard error of the mean, dependends on whether the population is finite or infinite. However, if the sample size is small relative to the population size, the two formulas are nearly identical. The form, or shape, of the sampling distribution of the sample mean depends on whether the population is normally distributed or not. If the population is normally distributed, the sampling distribution will be normally distributed. If the population is not normally distributed, the central limit theorem must be used.