Answer
$X^2\lt X_α^2$: the null hypothesis is not rejected.
Work Step by Step
$H_0:$ the number of students in the top 20% of the class is the same for each group.
$H_1:$ the number of students in the top 20% of the class is not the same for each group.
Total: top 20% $=0.20\times400=80$ students.
Expected values:
$E_1=E_2=E_3=E_4=0.20\times100=20$
$X^2=Σ\frac{(O_i-E_i)^2}{E_1}=\frac{(25-20)^2}{20}+\frac{(20-20)^2}{20}+\frac{(15-20)^2}{20}+\frac{(19-20)^2}{20}=2.55$
$k=4$. So, $d.f.=4-1=3$
$X_α^2=X_{0.05}^2=7.815$
(According to Table VII, for d.f. = 3 and area to the right of critical value = 0.05)
Since $X^2\lt X_α^2$, we do not reject the null hypothesis.
