Question

At BU, the mathematics department decided to offer a redesigned course called Network Analysis. The instructor wanted to determine if the grade distribution in the course differed from that of traditional courses. The follow shows the grade distribution of traditional courses based on historical records and the observed grades in the class of 80 students. A 0.15 B 0.35 С 0.25 F 0.05 0.2 Traditional Distribution Observed Counts 10 25 22 18 5 Does the sample evidence suggest that the distribution of grades is different from the traditional classes at the a = 0.05 level of significance? Hint: consider the hypotheses Ho: the grade distributions are the same versus Hy: the grade distributions are not the same. O a. Fail to reject Ho. There is sufficient evidence to conclude that the two grade distributions are different. b. Fail to reject Ho. There is not sufficient evidence to conclude that the two grade distributions are different. oc. Reject Ho. There is not sufficient evidence to conclude that the two grade distributions are different. O d. Reject Ho. There is sufficient evidence to conclude that the two grade distributions are different.

Answer 1

We have to perform Chi-square test for goodness of fit.

We have to test for null hypothesis

Ho : The grade distributions are the same

against the alternative hypothesis

H1 : The grade distributions are not the same

Our Chi-square test statistic is given by

$\tiny \chi^2=\sum_{i=1}^{n}\frac{\left(f_{i}-e_{i}\right)^2}{e_{i}}$

Here,

Number of grade classes

n = 5

Necessary calculations are as follows.

Grade Observed Expected frequency (f) proportion A 10 0.15 B 25 0.35 с 22 0.25 D 18 0.20 F 5 0.05 Total 80 1.00 Expected (f-e)^2/e frequency 12 0.3333 28 0.3214 20 0.2000 16 0.2500 4 0.2500 80 1.3548

$\tiny \therefore \chi^2_{calculated}=\sum_{i=1}^{5}\frac{\left(f_{i}-e_{i}\right)^2}{e_{i}}=1.3548$

Degrees of freedom $\tiny v=n-1=4$

$\tiny \therefore \text{p-value}=P\left(\chi^2_{4}>1.3548\right)=0.8520117$ [Using R-code '1-pchisq(1.3548,4)']

Level of significance $\tiny \alpha=0.05$

We reject our null hypothesis if $\tiny \text{p-value}<\alpha$

Here, we observe that $\tiny \text{p-value}=0.8520117\nless0.05=\alpha$

So, we cannot reject our null hypothesis.

Thus based on the given data we can conclude that there is no significant evidence that the grade distributions are not the same.

Hence, b. Fail to reject $\tiny H_0$ . There is not sufficient evidence to conclude that the two grade distributions are different.