Chi-SquareGoodness of Fit Test

Chi- Square Goodness of Fit test

• This test is applied when you have qualitative data from one single population

• Used to determine whether sample data is consistent with the hypothesized distribution– Example: If the M&M CO. claimed that 30% of M&M’s

were red, 40% green, 10% brown, 10% blue and 10% yellow, we could gather random samples of M&M’s and determine whether our distribution was different than the claim made from the company

Conditions

• Simple Random Sample• The population size is at least 10 times greater

than the sample size• The variable of the study is qualitative• The expected value of the sample

observations in each level of the variable is at least 5.

Conducting a Hypothesis Test1- State Hypothesis: Ho & Ha must be mutually exclusive

Ho: data consistent with a particular distributionHa: data that are not consistent with a particular distribution

* usually Ho specifies the proportion of observations at each level of the variable. The alternative hypothesis states that at least one of the specific proportions is not true

2- Formulate an Analysis Plan- describes how the sample data is used towards the null hypothesis

* use the chi square goodness of fit test. It is used to determine if the observed frequencies differ significantly from the expected frequencies stated in the null hypothesis.

3- Analyze: using sample data, find the degrees of freedom(DF),

expected frequency counts, test statistics, and P-value corresponding to the TS)

* DF= k-1 k= # of levels of the qualitative variable*Expected Frequency counts:

E= np E= expected frequency counts n= sample sizep= hypothesized proportion from the null

hypothesis *TS: Chi- Square random variable:

O= observed frequency count E= expected frequency count

*P- value- probability of observing the sample statistic as extreme as the TS where the TS is a chi square with degrees of freedom)

(can be found using Table C)

4- Interpret: Given the null hypothesis, if the sample results are unlikely, then reject the null. (done by comparing the P-value to the significance level and rejecting the null hypothesis if the P-value is less than the significant level)