Objectives

Test the difference between two large sample means using the z test.

Test the difference between two variances or standard deviations.

Objectives (cont'd.)

Test the difference between two means for small independent samples.

Test the difference between two means for small dependent samples.

Test the difference between two proportions.

The Difference Between Two Means

Assumptions for the test to determine the difference between two means:

The samples must be independent of each other; that is, there can be no relationship between the subjects in each sample.

The populations from which the samples were obtained must be normally distributed, and the standard deviations of the variable must be known, or the sample sizes must be greater than or equal to 30.

Formula

Formula for the z test for comparing two means from independent populations

General Formula Format

Observed difference

Expected difference

Standard error of difference

Difference Between Two Means

For Large Samples:

When n1" 30 and n2 " 30, and can be used in place of and .

F Distribution

If two independent samples are selected from two normally distributed populations in which the variances are equal ( ) and if the variances are compared as ,

the sampling distribution of the ratio of the variances is called the F distribution.

Characteristics of the F Distribution

The values of F cannot be negative, because variances are always positive or zero.

The distribution is positively skewed.

The mean value of F is approximately equal to 1.

The F distribution is a family of curves based on the degrees of freedom of the variance of the numerator and the degrees of freedom of the variance of the denominator.

Formula

The F test:

where is the larger of the two variances.

The F test has two terms for the degrees of freedoms: that...