Question

Grades and AM/PM Section of Stats: There were two large sections of statistics this term at State College, an 8:00 (AM) section and a 1:30 (PM) section. The final grades for both sections are summarized in the contingency table below.

Observed Frequencies: O_i's

	A	B	C	D	F	Totals
AM	6	11	17	19	18	71
PM	19	20	17	12	9	77
Totals	25	31	34	31	27	148

The Test: Test for a significant dependent relationship between grades and the section of the course. Conduct this test at the 0.05 significance level.

(a) What is the null hypothesis for this test?

H₀: The section (AM/PM) of a course and the grades are dependent variables.H₀: The section (AM/PM) of a course and the grades are independent variables.     

(b) What is the value of the test statistic? Round to 3 decimal places unless your software automatically rounds to 2 decimal places.

χ²

=

(c) Use software to get the P-value of the test statistic. Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value =

(d) What is the conclusion regarding the null hypothesis?

reject H₀fail to reject H₀    

(e) Choose the appropriate concluding statement.

We have proven that grades and section of the course are independent.The evidence suggests that there is a significant dependent relationship between grades and the section of the course.    There is not enough evidence to conclude that there is a significant dependent relationship between grades and the section of the course.

Answer 1

Expected	A	B	C	D	F
AM	25*71/148=11.99	31*71/148=14.87	34*71/148=16.31	31*71/148=14.87	27*71/148=12.95
PM	25*77/148=13.01	31*77/148=16.13	34*77/148=17.69	31*77/148=16.13	27*77/148=14.05

SD	A	B	C	D	F
AM	(11.99-6)^2/11.99=2.993	(14.87-11)^2/14.87=1.007	(16.31-17)^2/16.31=0.029	(14.87-19)^2/14.87=1.147	(12.95-18)^2/12.95=1.969
PM	(13.01-19)^2/13.01=2.758	(16.13-20)^2/16.13=0.929	(17.69-17)^2/17.69=0.027	(16.13-12)^2/16.13=1.057	(14.05-9)^2/14.05=1.815

b)
Test Statistic
The Chi-square test statistic is computed as follows
χ^2 = 2.993 + 1.007 + 0.029 + 1.147 + 1.969 + 2.758 + 0.929 + 0.027 + 1.057 + 1.815
χ^2 = 13.731

c)
The corresponding p-value of the test is p = P(χ^2 > 13.731) = 0.0082

d)
Reject H0

e)
The evidence suggests that there is a significant dependent relationship between grades and the section of the course.