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	19	18	17	71
PM	19	20	18	12	8	77
Totals	25	31	37	30	25	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 independent variables. H₀: The section (AM/PM) of a course and the grades are dependent 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

(a) The null alternative hypothesis for this test.

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

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

b)

The test statistic is

O: Observed frequency
E: Expected frequency.
E = ( Row total*Column total) / Grand total

O	E	(O-E)	(O-E)^2	(O-E)^2/E
6	11.9932432	-5.99324	35.91896	2.994933
11	14.8716216	-3.87162	14.98945	1.007923
19	17.75	1.25	1.5625	0.088028
18	14.3918919	3.608108	13.01844	0.904568
17	11.9932432	5.006757	25.06761	2.090145
19	13.0067568	5.993243	35.91896	2.761562
20	16.1283784	3.871622	14.98945	0.929384
18	19.25	-1.25	1.5625	0.081169
12	15.6081081	-3.60811	13.01844	0.834082
8	13.0067568	-5.00676	25.06761	1.927276
			Total	13.62

c)

Degrees of freedom = ( Number of rows - 1) * ( Number of column - 1) = (2 - 1) * (5 - 1) = 1 * 4 = 4

P-value = P() = 0.0086

d)

Level of significance = 0.05

P-value < 0.05 we reject null hypothesis.

reject H₀

e)

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