Question

500 students selected at random from 1500 students enrolled in a computer crash
programme were classified according to the age and grade points giving the
following data:
Age (in years)
Grade Point Below 20 21-30 Above 30
Up to 5 30 50 20
5.1 to 7.5 80 70 50
7.6 to 10.0 40 80 80

Test at 5% level of significance that age and grade points are independent

Answer 1

First we find the sum of each column and each row. Then we find the grand sum which is equal to 500.

$\text{Table 1:}\\ u_i=\frac{1}{500}\sum_{j=1}^c O_{ij}\text{ where } c\text{ --- number of columns},\\ O_{ij} \text{ --- observed values of the contingency table},\\ i=1, \ldots, r, r\text{ --- number of rows}.\\ \text{In our case } r=c=3.\\ v_j=\frac{1}{500}\sum_{i=1}^r O_{ij}\\ \text{Table 2:}\\ E_{ij}=500u_iv_j\text{ --- values of the table (expected values)}\\ \text{Table 3:}\\ a_{ij}=\frac{(E_{ij}-O_{ij})^2}{E_{ij}}\\ chi^2_o=\sum_{i=1}^r\sum_{j=1}^c a_{ij}=28.75\text{ --- observed value of } \chi^2\\ chi^2_{cr}\approx 9.49\text{ --- critical value of } \chi^2\\ chi^2_{cr}=chi^2_{cr}(alpha;df)\\ alpha=0.05 \text{ --- significance level}\\ df=(c-1)(r-1)=4\text{ --- degrees of freedom}\\ chi^2_o>chi^2_{cr}.\text{ So we can say that age and grade points are}\\ \text{not independent}.\\ \text{Using CHISQ.TEST() we find p-value. We can see}\\ \text{that p-value<alpha. So we can say that age and grade points are}\\ \text{not independent}.$