Chapter 7 Sampling Distributions

Frame

A list of elements thst the sample will be selected from

Standard error (of the mean)

Refers to the standard deviation of a point estimator. The value of this is helpful in determining how far the sample mean may be from the population mean. this is computed by using the (population standard deviation over the square root of the sample population).

Population Mean

The average of the population

Standard deviation of the sample mean

the standard deviation of xbar equals to the population standard deviation over the square root of the sample size, but only if the population is infinite or the population is finite and the sample size is less than or equal to 5% of the population size. If it is finite and is more than 5% of the population size its the (square root of the (total population minus the sample population) over the (total population minus one)) times the (population standard deviation over the square root of the sample population). Can provide probability information about the difference between the sample mean and the population mean.

Point Estimator

x bar, s, and p bar; the actual values are the point estimate but these values represent an estimation of what the populations values should be given room for error.

Sampled Population

The population from which the sample is drawn.

Standard deviation of pbar

With a finite population this equals to (the square root of (the population minus the sample population) over (the population minus 1)) multiplied by the (square root of (the population proportion times (1 minus the proportion) over the sample size). if it is infinite it is just (the population proportion times (1 minus the proportion) over the sample size).

Population Proportion

p, is the proportion of the population that fulfills a certain criteria. (success).

Sampling Distribution

A probability distribution consisting of all possible values of a sample statistic. Ex: all possible x bars. Knowledge of this sampling distribution and its properties will enable us to make probability statements about how close the sample mean is to the population mean. This distribution has a bell shaped distribution.

Random Sample

A sample of size n from an infinite population is a sample selected such that the following conditions are satisfied: 1) Each element selected comes from the same population. 2) Each element is selected independently.

Expected value of the sample mean

The mean of the sample mean random variable is the expected value of xbar. The expected value of xbar is the population mean.

Central Limit Theorem

In selecting random samples of size n from a population, the sampling distribution of the sample mean xbar can be approximated by a normal distribution as the sample size becomes large. If x has a mean of mu and a variance of sigma squared or it has enough observations (n>40) then theoretically it has a normal distribution.

Variance

Measures the uncertainty of the data. The bigger the sample size the less uncertainty it has. (less distance results will occur far away from the mean)

Parameters

Numerical Characteristics of a population

Simple Random Sample

Simplest type of probability sample in which each sample of size n from a finite population of size N has the same probability of being selected. Based upon the use of random numbers.

Population Standard Deviation

Square root of the population variance

Expected value of pbar

The expected value of pbar, the mean of all possible values of pbar, is equal to the population proportion p.

Sample proportion of pbar

The point estimator of the population proportion. pbar = x/n where x is the number of elements in the sample that posses the characteristics of interest and n being the sample size.

Sampling Distribution of x bar

The probability distribution of all possible values of the sample mean. This has an expected value or mean, a standard deviation and a characteristic shape or form.

Sampling Distribution of Pbar

The probability distribution of all possible values of the sample proportion of pbar. By understanding this we can determine how close the sample proportion pbar is to the population proportion. Can be approximated to have a normal distribution whenever np is >/= 5 or n(1-p) >/= 5. Can be used to provide probability information

Unbiased

When the expected value of a point estimator equals the population parameter.