Question

The mayor of a large city claims that 25 % of the families in the city earn more than $ 100,000 per year; 55 % earn between $ 30,000 and $ 100,000 (inclusive); 20 % earn less than $ 30,000 per year.

In order to test the mayor’s claim, 285 families from the city are surveyed and it is found that:

90 of the families earn more than $ 100,000 per year;
135 of the families earn between $ 30,000 and $ 100,000 per year (inclusive);
60 of the families earn less $ 30,000.

Test the mayor’s claim based on 2.5 % significance level.

Answer 1

Chi square test for Goodness of fit  
  

Ho: p1 = 0.25,p2 = 0.55 ,p3 = 0.20

H1: at least one proportion is different from given distribution

expected frequncy,E = expected proportions*total frequency  
total frequency=   285

category	observed frequencey, O	expected proportion	expected frequency,E			(O-E)²/E
>100,000	90	0.250	71.25			4.934
30,000<x<100,000	135	0.550	156.75			3.018
<30,000	60	0.200	57.00			0.158

chi square test statistic,X² = Σ(O-E)²/E =   8.110              
                  
level of significance, α=   0.025              
Degree of freedom=k-1=   3   -   1   =   2
                  
P value =   0.0173   [ excel function: =chisq.dist.rt(test-stat,df) ]          
Decision: P value < α, Reject Ho