Chi-square

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Chi-square – Goodness Of Fit Test

Purpose

The Chi-square Goodness of Fit Test is used to determine whether the observed frequencies in a sample could occur by chance when sampling from a population with an assumed distribution.

Anatomy


Reference: Black Book

Terminology

A. – ith category of the response
B. – observed frequency of response for category i
C. – expected frequency of response for category i
D. – The Chi-Square statistic
E. – Chi-Square critical to be compared with D
F. - The level of statistical error associated with a decision based on the statistic
G.– The degrees of freedom = (# of categories – 1)
H. – The number of response categories


Major Considerations

The sample size must be large enough to have each cell populated by 5 or more observations. The population must be sampled at random.

Application Cookbook

1. Sample from the population to obtain the observed frequencies. e.g. roll a die n times and count 1s, 2s, 3s, etc.
2. Enter the expected frequencies (e.g. for a die Fi = x n).
3. Calculate the Chi-square statistic.
4. Determine Chi-square critical and compare to the calculated value.
5. If Chi-square crit > Chi-square calc then the observed distribution of frequencies could have occurred purely by chance sampling from the assumed distribution. (The die is not biased.)

Chi-Square – Test Of Homogeneity


Purpose

To compare observed and expected frequencies of occurrence in a contingency table to test for independence of the variables.

Anatomy


Reference: Black Book

Terminology

A. Variable A
B. Variable B
C. k – Number of columns (equal to number of levels of Variable A)
D. r – Number of rows (equal to number of levels of Variable B)
E. fij – observed frequency of joint occurrence of variables
F. Fij – expected frequency of joint occurrence of variables
G. Li – Total of all occurrences of Bi
H. Cj – Total of all occurrences of Aj
I. n – Total number of observations
J. d.f. – Degrees of Freedom


Major Considerations

The population must be sampled at random. The sample size must be large enough for the expected frequency of each cell to be 5 or greater.
The variables are assumed to be independent.

Application Cookbook

1. Cross tabulate variables and create the contingency table per the Crosstabulation and Contingency Table tool.
2. Calculate the Chi-square statistic.
3. Determine the degrees of freedom.
4. Choose alpha.
5. Determine Chi-square crit from tables or Excel using CHINV and the appropriate alpha and degrees of freedom.
6. If Chi-square calculated > Chi-square critical accept Ho i.e. the variables are independent with a confidence level equal to (1 - alpha). Otherwise – reject Ho, and the variables cannot be considered independent.
7. An alternative to the manual calculation is to enter the table in Minitab and use STAT - TABLES - Chi-Square Test


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