Chi-squared Distribution Statistical Inference for Managers

Chi-squared test The test we use to measure the differences between what is observed and what is expected according to an assumed hypothesis is called chi-squared test. This test enables us to see how well does the assumed theoretical distribution (e.g. normal distribution) fit to the observed data.

Chi-squared test Applications of chi-squared test: The test can be used in: Goodness of fit of distributions. Test of independence of attributes. Test of homogeneity.

Hypothesis Hypothesis: H0: No association H1: Association X²= Σ(O-E)²/ E O= Observed frequency E= Expected frequency fE= Expected frequency= (Column total)(row total) Grand total X²= Σ(f0-fe)²/ fe

Degree of freedom and contingency table Degree of freedom= (r-1)(c-1) r= no. of rows c= no. of columns Conclusion: If X² calculated˃ X² table with (n-1) degrees of freedom, then reject H0 and it can be concluded that there is significant association between the two attributes. Test enables us to explain whether or not two attributes are associated. Example: If we want to know whether a new medicine is effective in controlling fever or not using X² test.

Example Calculate the sample X² value? Contingency table North East South East Central West Coast Number who prefer present 68 75 57 79 method Prefer new 32 45 33 31 Calculate the sample X² value?

Formulas repeated! Formulas to remember: E= (row total)(column total)/ grand total X²= (O-E)²/ E v= (r-1)(c-1)

Example-2 To see whether silicon chip sales are independent of where the US economy is in the business cycle, data have been collected on the weekly sales of Zippy Chippy, a silicon valley firm and on whether the US economy was rising to a cycle peak, falling to a cycle trough or at a cycle trough. The results are: High Medium Low Economy At peak 20 7 3 At trough 30 40 30 Rising 20 8 2 Falling 30 5 5

Example-2 State the null and alternative hypothesis. Calculate the sample X² value. At the 10% significance level, what is your conclusion.