Chapter 7: Sampling and sampling distributions

Simple random sample (finite population)

A simple random sample of size, n, from a finite population of size, N, is a sample selected so that each possible sample size, n, has the same probability of being selected.

Standard deviation of P

Finite population: Sigma P = roten ur pi(1-pi) / n x roten ur (N-n)/(N-1) Infinite population: Sigma P = roten ur pi(1-pi)/n If the population is finite with n/N< 0,05 we use the infinite population formula

Standard deviation of the sample mean (X with a bar)

Finite population: Sigma X with bar = (sigma / roten ur n) x roten ur (N-n/N-1) The factor roten ur (N-n/N-1) is the finite population correction factor. Sigma X with bar = the standard error of the mean Infinite population Sigma X with bar = sigma / roten ur n

Sampling with or without replacement

If we select a sample and previously used numbers are acceptable we sample with replacement. If a random number is selected more than once and we ignore it, we sample without replacement.

Simple random sample (infinite population)

In some cases the population is very large so it is treated as infinite. For example bank transactions and customers entering stores. The random number selection cannot be used for infinite populations. A simple random sample from an infinite population is a sample selected so that the following conditions are met: 1. Each element selected comes from the population 2. Each element selected is selected independently of other elements.

Use the infinite population expression to compute the standard deviation of the sample mean IF:

Sigma X with bar = sigma /roten ur n When: 1. The population is infinite OR 2. The population is finite AND the sample size is less than or equal to 5% of the population size, that is n/N < or = 0,05

Expected value of P

The expected value of P, the mean of all possible values of P, is equal to the population proportion, pi. P is an unbiased estimator of pi. E (P) = pi E (P) = the expected value of P Pi = the population proportion

Point estimator and point estimate

The numerical value you get for the sample mean, sample standard deviation or sample proportion is called the point estimate. The point estimate helps with calculating a value from a sample statistic to use as a population parameter. A population parameter is a numerical descriptive measure of a population. We refer to the sample mean (X with a bar) as the point estimator of the population mean, mu. S (sample standard deviation) is the point estimator of the population standard deviation, sigma. P (sample proportion) is the point estimator of the population proportion, pi.

Expected value of the sample mean (X with a bar)

The sample mean values given by the various possible simple random samples. The mean of all the values is the expected value of the sample mean. E (X with a bar) = mu Mu = the mean of the population which the sample is selected E (X with a bar)= the expected value of the sample mean

Form of the sampling distribution of P

The sample proportion is p = m/n. For a simple random sample from a large population, the value m is a binomial random variable that indicates the number of elements in the sample. n is constant. The sampling distribution of P can be approximated by a normal distribution whenever npi > 5 and n(1-pi) > 5

Sampling distribution of P

The sampling distribution of P is the probability distribution of all possible values of the sample proportion, P. To determine how close the sample proportion is to the population proportion, pi. We need to understand the properties of the sampling distribution of P.

Sampling distribution of the mean (X with a bar)

The sampling distribution of the sample mean is the probability distribution of all of the means possible values.

Sampling error and unbiasedness

When the expected value of a point estimator equals the population parameter, the point estimator is unbiased. The sample statistic Q is an unbiased estimator of the population parameter 0 med streck if E(Q) = 0 med streck. Where E(Q) is the expected value of the sample statistic Q. The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. The error happens because a sample and not the entire population is used to estimate a population parameter. Mean - mu S - sigma P - pi