Question

~Please provide explanations of your findings and its business implications

You have been asked to ascertain whether there is a relationship between the geographic region of the country in which the private universities are located and the categorical usual amount of financial aid awarded yearly to students. Sample data concerning these two variables is given in appendix three. The numbers in the table represent the numbers of universities in each category that regularly award their students that categorical amount of financial aid. At both the 5% and 10% levels of significance, is there evidence of a statistical relationship between the location of the university and the usual amount of financial aid offered students by the university?

Appendix Three:

                                                                                    Region

Categorical Amount            North              South             East                West

<$5,000                                  12                    9                     15                    6

$5,000-$9,999                      15                    11                    20                    12

$10,000-$14,999                  17                    16                    19                    10

$15,000-$19,999                  25                    18                    23                    11

>$19,999                               21                    10                    29                    8

Answer 1

We use a chi square test of independence to check if there is a statistical relationship between the 2 variables.

The Expected value data are in the table below. Each Cell = Row total * Column Total/N. N = 307

Observed
	North	South	East	West	Total
< $5000	12	9	15	6	42
$5000 - $9,999	15	11	20	12	58
$10,000 - $14,999	17	16	19	10	62
$15,000 - $19,999	25	18	23	11	77
> $19,999	21	10	29	8	68
Total	90	64	106	47	307
Expected
	North	South	East	West	Total
< $5000	12.31	8.76	14.50	6.43	42
$5000 - $9,999	17.00	12.09	20.03	8.88	58
$10,000 - $14,999	18.18	12.93	21.41	9.49	62
$15,000 - $19,999	22.57	16.05	26.59	11.79	77
Master	19.93	14.18	23.48	10.41	68
Total	90.00	64.00	106.00	47.00	307

The degrees of freedom, df = (r – 1) * (c -1) = (5 - 1) * (4 - 1) = 4 * 3 = 12

______________________________________________

The Hypothesis:

H0: There is no relation between Location of the University and the Financial Aid offered.

Ha: There is a relation between Location of the University and the Financial Aid offered.

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The Test Statistic: The table below gives the calculation of .

	Observed	Expected	O-E	(O-E)2	(O-E)2/E
1	12	12.31	-0.31	0.10	0.01
2	15	17.00	-2.00	4.01	0.24
3	17	18.18	-1.18	1.38	0.08
4	25	22.57	2.43	5.89	0.26
5	9	8.76	0.24	0.06	0.01
6	11	12.09	-1.09	1.19	0.10
7	16	12.93	3.07	9.46	0.73
8	18	16.05	1.95	3.79	0.24
9	15	14.50	0.50	0.25	0.02
10	20	20.03	-0.03	0.00	0.00
11	19	21.41	-2.41	5.79	0.27
12	23	26.59	-3.59	12.86	0.48
13	6	6.43	-0.43	0.18	0.03
14	12	8.88	3.12	9.74	1.10
15	10	9.49	0.51	0.26	0.03
16	11	11.79	-0.79	0.62	0.05
17	21	19.93	1.07	1.13	0.06
18	10	14.18	-4.18	17.44	1.23
19	29	23.48	5.52	30.48	1.30
20	8	10.41	-2.41	5.81	0.56
Total					6.77451

test = 6.77

______________________

The Critical Value:

critical (0.05) = 21.03

critical (0.10) = 18.55

The p value: The p value: The p value at test = 6.77, df = 12; P value = 0.8724

__________________________

The Decision Rule: If test is > critical, then Reject H0.

If p value is < , Then Reject H0.

___________________________

The Decision:   

At = 0.05

Since test (6.77) is < critical (21.03), We Fail to reject H0.

Since p value (0.8724) is > (0.05), We Fail To Reject H0.

___

At = 0.10

Since test (6.77) is < critical (18.55), We Fail to reject H0.

Since p value (0.8724) is > (0.10), We Fail To Reject H0.

___________________________________

The Conclusion (at 5%): There isn't sufficient evidence at the the 95% significance level to conclude that there is relationship between Location of the University and the Financial Aid offered.

The Conclusion (at 10%): There isn't sufficient evidence at the the 90% significance level to conclude that there is relationship between Location of the University and the Financial Aid offered.

_____________________________________