Because “Male” and “Female” are now considered separate populations, we do not ask if gender and
political party preference are independent in the population of students. We ask instead if the proportions
of Democrats, Republicans, and Independents are the same within the populations of Males and Females.
This is the test for homogeneity of proportions (or homogeneity of populations). Let the proportion of
Male Democrats be p 1 ; the proportion of Female Democrats be p 2 ; the proportion of Male Republicans
be p 3 ; the proportion of Female Republicans be p 4 ; the proportion of Independent Males be p 5 ; and the
proportion of Independent Females be p 6 . Our null and alternative hypotheses are then
H 0 : p 1 = p 2 , p 3 = p 4 , p 5 = p 6 .H (^) A : Not all of the proportions stated in the null hypothesis are true.
It works just as well, and might be a bit easier, to state the hypotheses as follows.
H 0 : The proportions of Democrats, Republicans, and Independents are the same among Male and
Female students.
H (^) A : Not all of the proportions stated in the null hypothesis are equal.
For a given two-way table the expected values are the same under a hypothesis of homogeneity or
independence.
example: A university dean suspects that there is a difference between how tenured and
nontenured professors view a proposed salary increase. She randomly selects 20 nontenured
instructors and 25 tenured staff to see if there is a difference. She gets the following results.
Do these data provide good statistical evidence that tenured and nontenured faculty differ in their attitudes
toward the proposed salary increase?
solution:
I. Let p 1 = the proportion of tenured faculty who favor the plan and let p 2 = the proportion of
nontenured faculty who favor the plan.
H 0 : p 1 = p 2 .
H (^) A : p 1 ≠ p 2 .