Answer
$X^2\gt X_α^2$: null hypothesis is rejected.
There is enough evidence to conclude that hockey players’ birthdates are not uniformly distributed throughout the year.
Work Step by Step
$H_0:$ hockey players’ birthdates are uniformly distributed throughout the year.
That is: $P(January–March)=P(April–June)=P(July–September)=P(October–December)=\frac{1}{4}=0.25$
$H_1:$ hockey players’ birthdates are not uniformly distributed throughout the year.
Total: $63+56+28+34=181$ players.
Expected count of January–March: $181\times0.25=45.25$
Expected count of April–June: $181\times0.25=45.25$
Expected count of July–September: $181\times0.25=45.25$
Expected count of October–December: $181\times0.25=45.25$
$X^2=Σ\frac{(O_i-E_i)^2}{E_1}=\frac{(63-45.25)^2}{45.25}+\frac{(56-45.25)^2}{45.25}+\frac{(28-45.25)^2}{45.25}+\frac{(34-45.25)^2}{45.25}=18.89$
$k=4$. 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 $X^2\gt X_α^2$, we reject the null hypothesis.
