## Glimpses into Application of Chi-Square Tests in Marketing

| May 18,2010 12:01 pm IST

In this article, an attempt is made to bring into sharp focus the use of c2 in marketing function. By no means, the coverage is exhaustive.

The aim is to make the reader appreciate the conceptual framework of Chi-Square analysis through problem illustrations in marketing. The ideas presented in this article certainly can be extended to many decision situations in marketing that can fruitfully employ chi-square tests.

Contents:

**1. Chi-Square Analysis - Introduction2. Chi-Square Test-Goodness of Fit3. Chi-Square Test of Independence4. Solution5. Calculation**

**1. Chi-Square (c2 ) Analysis - Introduction**

Consider the following decision situations:

1) Are all package designs equally preferred?

2) Are all brands equally preferred?

3) 3) Is their any association between income level and brand preference?

4) Is their any association between family size and size of washing machine bought?

5) Are the attributes educational background and type of job chosen independent?

The answer to these questions requires the help of Chi-Square (c2 ) analysis. The first two questions can be unfolded using Chi-Square test of goodness of fit for a single variable while solution to questions 3, 4, and 5 need the help of Chi-Square test of independence in a contingency table. Please note that the variables involved in Chi-Square analysis are nominally scaled. Nominal data are also known by two names - categorical data and attribute data.

The symbol c2 used here is to denote the chi-square distribution whose value depends upon the number of degrees of freedom (d.f.). As we know, chi-square distribution is a skewed distribution particularly with smaller d.f. As the sample size, and therefore, the d.f. increases and becomes large, the c2 distribution approaches normality. c2 tests are nonparametric or distribution-free in nature. This means that no assumption needs to be made about the form of the original population distribution from which the samples are drawn. Please note that all parametric tests make the assumption that the samples are drawn from a specified or assumed distribution such as the normal distribution.

__Conditions for using the c2 Test__

The sample observations drawn from a population must be independent and random.

The data must be in frequency (counting) form. If the original data are in percentages, they must be converted into frequency.

No frequency in any cell/category must be less than 5. If the frequency is less than 5 for a category, you have to do some regrouping.

**2. Chi-Square Test-Goodness of Fit**

A number of marketing problems involve decision situations in which it is important for a marketing manager to know whether the pattern of frequencies that are observed fit well with the expected ones. The appropriate test is the c2 test of goodness of fit. The illustration given below will clarify the role of c2

Problem: In consumer marketing, a common problem that any marketing manager faces is the selection of appropriate colors for package design. Assume that a marketing manager wishes to compare five different colors of package design. He is interested in knowing which of the five is the most preferred one so that it can be introduced in the market. A random sample of 400 consumers reveals the following:

Do the consumer preferences for package colors show any significant difference?

Solution: If you look at the data, you may be tempted to infer that Blue is the most preferred color. Statistically, you have to find out whether this preference could have arisen due to chance. The appropriate test statistic is the c2 test of goodness of fit.

Null Hypothesis: All colors are equally preferred.

Alternative Hypothesis: They are not equally preferred.

Please note that under the null hypothesis of equal preference for all colors being true, the expected frequencies for all the colors will be equal to 80. Applying the formula,

we get the computed value of chi-square ( c2 ) = 11.400. The critical value of c2 at 5% level of significance for 4 degrees of freedom is 9.488. So, the null hypothesis is rejected. The inference is that all colors are not equally preferred by the consumers. In particular, Blue is the most preferred one. The marketing manager can introduce blue color package in the market.

**3. Chi-Square Test of Independence**

The goodness-of-fit test discussed above is appropriate for situations that involve one categorical variable. If there are two categorical variables, and our interest is to examine whether these two variables are associated with each other, the chi-square ( c2 ) ) test of independence is the correct tool to use. This test is very popular in analyzing cross-tabulations in which an investigator is keen to find out whether the two attributes of interest have any relationship with each other. .