Question

Traditionally 35% of the students at Wortham University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College. To see whether or not the proportions have changed, a sample of 300 students was taken. Ninety of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are in the Education College. The hypothesis is to be tested at the 5% level of significance. The critical value from the table equals

Answer 1

The following table is obtained:

Categories	Observed	Expected	(fo-fe)2/fe
Category 1	90	300*0.35=105	(90-105)2/105 = 2.143
Category 2	120	300*0.35=105	(120-105)2/105 = 2.143
Category 3	90	300*0.30=90	(90-90)2/90 = 0
Sum =	300	300	4.286

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H0​:p1​=0.35,p2​=0.35,p3​=0.30

Ha​:  the proportions have changed,

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, the number of degrees of freedom is df=3−1=2, critical value = 5.991

so then the rejection region for this test is R={χ2:χ2 > 5.991}.

(3) Test Statistics

The Chi-Squared statistic is computed as follows:

χ2 = ∑​(Oi​−Ei​)² / Ei ​

= 2.143+2.143+0

=4.286

Decision about the null hypothesis

Since it is observed thatχ2=4.286 ≤χc2 ​=5.991, it is then concluded that the null hypothesis is not rejected.

Conclusion

It is concluded that the null hypothesis Ho is not rejected ,there is NOT enough evidence to claim that

proportions have changed,