Test because that Homogeneity

Summary: This module defines how the chi-square distribution can be used to test because that homogeneity.

You are watching: Why does the test for homogeneity follow the same procedures as the test for independence?

The kindness of right test deserve to be provided to decision whether a populace fits a provided distribution, yet the quality of right test will certainly not suffice to to compare whether two populaces follow the same unknown distribution. A various test, dubbed the Test because that Homogeneity, have the right to be supplied to do a conclusion about whether two populations have actually the same distribution. To calculation the test statistic because that a test because that homogeneity, follow the same procedure similar to the c2 test because that independence.

Here is a an introduction of the Test because that Homogeneity:

Hypotheses

H0: The distributions of the two populations are the same.Ha: The distribution of the two populaces are no the same.

Test Statistic

Uses a c2 statistic. That is computed in the same method as the test for independence.

Requirements

All values in the table need to be higher than or same to 5.

Common Uses

Comparing 2 populations. For example: men vs women, prior to vs. After, east vs. West.

The variable is categorical with more than two possible response values.

EXAMPLE 1

Do male and female university students have actually the same distribution of living conditions? usage a level of significance of 0.05. Expect that 250 randomly selected male college students and also 300 randomly selected female college students to be asked about their life conditions: Dorm, Apartment, with Parents, Other. The outcomes are presented in the table below:

DormApartmentWith ParentsOtherMale72844945Female91868835

Solution

The null and alternate hypotheses are:
H0:The distribution of living conditions for masculine college student is the exact same as the circulation of living problems for female university students.
Ha:The circulation of living problems for masculine college student is the no the exact same as the distribution of living problems for female college students.

To compute the check statistic, follow the same procedure as with the test because that independence. Here there space 2 rows and also 4 columns. Keep in mind that the degrees of flexibility for this test is df = the number of columns - 1 = 3

To the appropriate is the readout indigenous a TI 84+.

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The c2 test statistic is about 10.13 and the p-value is 0.0175. As with all theory tests, reject the null hypothesis if the p-value is less than the level the significance and fail to reject the null theory if the p-value is better than the level that significance. In this case,

p-value = 0.0175

Therefore, reject the null hypothesis and accept the alternate hypothesis. You have the right to conclude that the distribution of living problems for male and also female university students are not the same.

Notice that the conclusion is just that the distributions are not the same. One cannot usage the test because that homogeneity come make any kind of conclusions around how they differ.

EXAMPLE 2

Both before and after a recent earthquake surveys were performed asking voters which of the 3 candidates lock planned ~ above voting for in the upcoming city board of directors election. Has actually there been a readjust since the earthquake? use a level of definition of 0.05. The table below shows the outcomes of the survey.

PerezChungStevens
Before167128135
After214197225

Solution

The null and different hypotheses are:
H0:The distribution of voter choices was the exact same before and also after the earthquake.
Ha:The distribution of voter preferences was no the exact same before and after the earthquake.

This table has actually 2 rows and also 3 columns. The degrees of freedom for this test is

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df = the variety of columns - 1 = 2

To the right is the readout indigenous a TI 84+. The c2 check statistic is about 3.26 and also the p-value is 0.196. The inequality is

p-value = 0.196 > 0.05 = Level that Significance

Therefore, failure to reject the null hypothesis. Over there is insufficient proof to do a conclusion about whether the distribution of voter preferences differs before and after the earthquake.

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Summary the a c2-Tests

You have actually seen the a c2 check statistic provided in three different circumstances. Below is a summary that will help you decide which c2 check is the proper one to use.

Goodness of Fit: use the goodness of fit Test when you want to decision whether a population with unknown circulation "fits" a well-known distribution. In this case there will be a solitary qualitative survey concern or a single outcome of one experiment from a solitary population. Kindness of fit is frequently used to check out if the population is uniform (all outcomes take place with equal frequency), the population is normal, or the population is the same as another populace with recognized distribution. The null and alternative hypotheses are:H0: The populace fits the given distribution.Ha: The population does no fit the provided distribution.Independence: usage the test for self-reliance when you desire to decision whether two variables room independent or dependent. In this situation there will certainly be two qualitative survey questions or experiments and also a contingency table will be constructed. The score is to check out if the 2 variables room unrelated (independent) or associated (dependent). The null and alternative hypotheses are:H0: The two variables are independent.Ha: The 2 variables room dependent.

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Homogeneity: use the Test for Homogeneity as soon as you desire to decision if two populaces with unknown circulation have the same circulation as each other. In this instance there will certainly be a solitary qualitative survey inquiry or experiment offered to two different populations. The null and alternate hypotheses are:H0: The two populaces follow the very same distribution.Ha: The two populaces have different distributions.