Question

A teacher is trying to assign grades so that she has the following distribution:

Grade	% of Class
A	15
B	30
C	40
D	10
F	5

This is what she actually has in her grade book:

Grade	# Students
A	20
B	40
C	35
D	8
F	8

At a significance level of 0.05, are these two distributions different? Include either χ2 or the p-value to justify your answer.

Answer 1

Total Frequency = 20 + 40 + 35 + 8 + 8 = 111

Therefore the expected frequencies for each grade here is computed as:
E(A) = 0.15*111 = 16.65
E(B) = 0.3*111 = 33.3
E(C) = 0.4*111 = 44.4
E(D) = 0.1*111 = 11.1
E(F) = 0.05*111 = 5.55

Therefore now the chi square test statistic here is computed as:

Grade	O_i	E_i	Chi-Sq
A	20	16.65	0.67402402
B	40	33.3	1.34804805
C	35	44.4	1.99009009
D	8	11.1	0.86576577
F	8	5.55	1.08153153
	111	111	5.95945946

From the above bold value, we get here:

For k - 1 = 4 degrees of freedom, the p-value here is obtained from the chi square distribution tables as:

As the p-value here is 0.2022 > 0.05 which is the level of significance, therefore the test is not significant and we cannot reject the null hypothesis here. Therefore we have insufficent evidence here that the distribution is different here.