This test involves with nominal data produced by
multinomial experiment It is a generalisation of a binomial experiment These test the null hypothesis that data in the target population has a particular probability distribution. Example 1 We might test whether consumers are indifferent to which of four materials (glass, plastic, steel or aluminium) that could be used to make soft drink containers.

i.e. there is only a 5 percent chance or less that 23 > 7.82 if HO is true.
Comparison of chi square values


23 = 12.08 > 7.82 reject HO.
Conclusion: at the 5% significance level there is sufficient evidence to reject the null hypothesis. At least one of the probabilities (pi) is different. The
Test Statistics Main Display Colour Degrees of freedom, groups - 1
Chi-Square
df Asymp. Sig.
a
2.467 2 .291
a. 0 cells (.0%) have expected frequencies less than 5. The minimum expected cell frequency is 30.0. Chi-square statistic
SPSS单样本非参数检验
总体分布的chi-square检验
(1)目的: 根据样本数据推断总体的分布与某个已知分布是否有显著差异--吻合性检验。
适用于分类资料的统计推断
SPSS单样本非参数检验

总体分布的chi-square检验
(2)基本假设: H0:总体分布与理论分布无显著差异 (3)基本方法 – 根据已知总体的构成比计算出样本中各类别的期望频 数，计算实际观察频数与期望频数的差距,即：计算卡

The null hypothesis is that they don’t have colour preference. The default is that the probabilities are equal.
Use Analyse/Nonparametric tests /Chi-Square.
sample results indicate that the materials are not equally preferred by consumers in the target population. Thus, at least preferences for two
materials are different.
two categorical variables
Gender and preference for a product, whether the
preference for a product is independent from gender
Chi-square test for differences between proportions
Ho: Consumers in the target population have no preference for any of three colours of packaging H1: Consumers in the target population have preference for at least one of three colours of packaging.

Example: We test the null hypothesis that consumers in the target population have no preference for any of three colours of packaging.
Main display colour Observed N 26 37 27 90 Expected N 30.0 30.0 30.0 Residual -4.0 7.0 -3.0
Blue Green Purple Total
Numbers of consumers actually choosing particular colours.
Numbers of consumers expected to choose particular colours if the null is true.

Categorical variable
Variables that describe categories of entities
Dealing with them all the time in statistics Making comparisons among variables For example, whether consumers prefer a particular brand of a product among other competing brands. Checking whetቤተ መጻሕፍቲ ባይዱer there is a relationship between
2 3 (39 25)2 (16 25)2 (20 25)2 (25 25)2 25 25 25 25
2 3 12 . 08
Obtain the critical value of chi square

Critical 23 = 7.82. Obtain the critical value at 5% significance level at 3 d.f., (Table E4, page 742, Berenson et.al. 2013)

The null hypothesis is that they are indifferent (or that equal numbers prefer glass, plastic, steel and aluminium).
Example 1
Data


Let pG be the probability that an individual selected at random will nominate glass as his/her preference if required to make a choice. Similarly for pP (plastic), pS (steel) and pA (aluminium) HO: pG = pP = pS = pA = 0.25. HA: at least one pi 0.25.
Chi square test using SPSS
Example : Suppose that we want to test whether or not
customers have a colour preference for packaging. Three different colours, Blue, Green & Purple, are considered.

Hypotheses


The alternative is that at least one material is more preferred (or less preferred) than the others.
Example 1cont..
Procedure:

Select a random sample of, say, 100 consumers and determine their preferences.
Week Six – Analyzing categorical data: Chi-squared tests
This week lecture will cover...
Analysing categorical data (nominal) Chi-square test of differences between proportions Chi-square test of independence
Main display colour Observed N 26 37 27 90 Expected N 30.0 30.0 30.0 Residual -4.0 7.0 -3.0
Blue Green Purple Total
Different but different enough to reject the null?
Under the null hypothesis We expect 25 consumers to nominate glass, 25 to nominate plastic, 25 to nominate steel and 25 to nominate aluminium

These are the expected frequencies, Ei.