Question

) A sample of 15 graduating students, 15 females and 5 males, was randomly selected. Their GPAs were recorded, and the ranks of these GPAs for the pooled sample are presented below. Test the following hypothesis: H₀: There is no difference in GPAs between the genders? Use the Mann-Whitney test.

Males	Females
11	1
12	2
3	13
14	4
6	5
	15
	7
	8
	9
	10

Calculated test statistic:

Critical value of the test statistic:

Decision rule:

Decision:

Answer 1

GIVEN:

Sample size of males

(m) = 5

Sample size of females

(no) = 10

HYPOTHESIS:

The hypothesis is given by,

$H_{0}:\theta _{1}=\theta _{2}$ (That is, there is no significant difference in GPA's between males and females.)

$H_{1}:\theta _{1}\neq \theta _{2}$ (That is, there is significant difference in GPA's between males and females.)

LEVEL OF SIGNIFICANCE:

Let us assume significance level to be .

a-0.05

TEST STATISTIC:

$U=min(U_{1},U_{2})$

where

$U_{1}=n_{1}n_{2}+[(n_{1}(n_{1}+1))/2]-\sum R_{1}$

$U_{2}=n_{1}n_{2}+[(n_{2}(n_{2}+1))/2]-\sum R_{2}$

CALCULATION:

Let us first arrange GPA's of males and females in ascending order separately.

Males	Females
3	1
6	2
11	4
12	5
14	7
	8
	9
	10
	13
	15

Now we should give combined ranks to both groups.

Males	$R_{1}$	Females	$R_{2}$
3	3	1	1
6	6	2	2
11	11	4	4
12	12	5	5
14	14	7	7
		8	8
		9	9
		10	10
		13	13
		15	15
	$\sum R_{1}=46$		$\sum R_{2}=74$

Now

$U_{1}=n_{1}n_{2}+[(n_{1}(n_{1}+1))/2]-\sum R_{1}$

$=(5*10)+[(5(5+1))/2]-46$

$=50+15-46$

01-19

$U_{2}=n_{1}n_{2}+[(n_{2}(n_{2}+1))/2]-\sum R_{2}$

$=(5*10)+[(10(10+1))/2]-74$

$=50+55-74$

$U_{2}=31$

$U=min(U_{1},U_{2})$

$=min(19,31)$

$U=19$

CRITICAL VALUE:

From the Mann Whitney U test critical value table, the two tailed (since $H_{1}:\theta _{1}\neq \theta _{2}$ ) critical value with $n_{1}=5$ and
n2 10
at significance level
a-0.05
is .

DECISION RULE:

$Reject \, \, H_{0}\, \, if\, \, U\leq 8$

CONCLUSION:

Since the calculated test statistic value (19) is greater than the critical value (8), we fail to reject null hypothesis and conclude that there is no significant difference in GPA's between males and females.