CHI-SQUARE TEST X2

Another most widely used test of significance ( non- parametric).Particularly useful in test involving cases where persons, events or objects are grouped in two or more nominal categories such as yes or no, approved-undecided disapprove or class “A,B,C,D.”

• By using the Chi-square , we can test for the significant differences between the observed distribution of data among categories and expected distribution of data based upon the null hypothesis.

• It is useful in case of one-sample analysis, two independent samples or k independent samples

1. One variable Chi-square( goodness- of –fit test)Compares a set of observed frequency (0) for each categories.( Example: political candidates, contestant 1,2,3)2. Two variable chi- square( test of independence)With two or more categories/ produce to determine whether two or more variables are statistically independent( Example: Classification of heights as first variable and weight as second variable)3. Test of Homogeneity the test is concerned with two or more samples, with only one criterion variable. It is used to determine if two or more populations are

homogeneous.

Forms of chi- square X2

• 1.Make a problem statement

• 2. Hypotheses

• H0:  The distributions of the two populations are the same.

•Ha:  The distributions of the two populations are not the same.

Solving by the step method

• The significance level, α corresponds to the size of the rejection region. It determines how small the p-value should be in order to reject the null hypothesis. The common choices for α are 0.05, 0.01, or 0.10.

Significant level

P-value• The p-value is the probability of

getting a value for the test statistic as large or larger than the observed value of the test statistic just by random chance.

• To determine a p-value look at the χ2 table with df degrees of freedom and find where the observed value of the χ2 statistic falls on this table.

Level of Significance:• Degrees of freedom.

• DF = (r - 1) (c - 1)

• where r is the number of populations/no. of rows, and

• c is the number of levels for the categorical variable/no. of columns.

• X2 = ∑

• Where:• X2 = the chi-square test• O = the observed frequencies• E = the expected frequencies

Use the formula for any form of contingency table

• there are two ways to make a decision in this test

• Classical Reject null hypothesis if χ2 ≥ χ2 α,df Fail to reject null hypothesis if χ2 < χ2 α,df OR

P-value: • this method is preferred by researchers

currently conducting research Reject null hypothesis if p-value ≤ α Fail to reject null hypothesis if p-value > α

The decision rule

• The conclusion is a statement written to convey the results of the research. If possible, avoid statistical terminology and should be written in a form that can be easily understood by non-statisticians.

• Example• If the null hypothesis is rejected then conclude that the two

variables are not homogeneous at the specified significance level.

• If the null hypothesis is not rejected then conclude that the variables are homogeneous at the specified significance level.

The conclusion