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.
Reference: Black Book
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
The sample size must be large enough to have each cell populated by 5 or more observations. The population must be sampled at random.
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.)
To compare observed and expected frequencies of occurrence in a contingency table to test for independence of the variables.
Reference: Black Book
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
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.
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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