Chapter 27, Problem 27.35E

To determine

To explain is there evidence of significant improvement within each group and did the treatment group show significantly greater improvement then the control group.

Answer to Problem 27.35E

Yes, there is evidence of significant improvement within each group but the treatment group showno significantly greater improvement then the control group.

Explanation of Solution

In the question, it is given the scores of the mathematical skills before and after a subliminal message for both the treatment group and control group. Now, to find that whether there evidence of significant improvement within each group, we will perform the Wilcoxon signed rank test for each group by the software. Thus, by performing the Wilcoxon signed rank test for the treatment group let us first find the difference and then the results for the test by software are as follows:

Null hypothesis: There is no difference between them.

Alternative hypothesis: There is evidence of significant improvement within group.

	Treatment
Before	After	Difference Treatment
18	24	-6
18	25	-7
21	33	-12
18	29	-11
18	33	-15
20	36	-16
23	34	-11
23	36	-13
21	34	-13
17	27	-10

Data	Rank
-6	1
-7	2
-12	6
-11	4.5
-15	9
-16	10
-11	4.5
-13	7.5
-13	7.5
-10	3

variables:	Before - After
0	sum of positive ranks
55	sum of negative ranks

10	n
27.50	expected value
9.50	standard deviation
-2.89	z, corrected for ties
.0019	p-value (one-tailed, lower)

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we have sufficient evidence to conclude that there is evidence of significant improvement within treatment group.

Thus, now by performing the Wilcoxon signed rank test for the control group let us first find the difference and then the results for the test by software are as follows:

Null hypothesis: There is no difference between them.

Alternative hypothesis: There is evidence of significant improvement within group.

Control
Before	After	Difference Control
18	29	-11
24	29	-5
20	24	-4
18	26	-8
24	38	-14
22	27	-5
15	22	-7
19	31	-12

Data	Rank
-11	6
-5	2.5
-4	1
-8	5
-14	8
-5	2.5
-7	4
-12	7

variables:	Before - After
0	sum of positive ranks
36	sum of negative ranks

8	n
18.00	expected value
6.93	standard deviation
-2.60	z, corrected for ties
.0047	p-value (one-tailed, lower)

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we have sufficient evidence to conclude that there is evidence of significant improvement within control group.

Now, to find did the treatment group show significantly greater improvement then the control group, we will use the Wilcoxon rank sum test by using the software. Thus, the hypotheses will be defined as:

Null hypothesis: The treatment group show no significant improvement then the control group.

Alternative hypothesis: The treatment group show significantly greater improvement then the control group.

Label	Data	Rank
Difference Treatment	-6	15
Difference Treatment	-7	13.5
Difference Treatment	-12	6.5
Difference Treatment	-11	9
Difference Treatment	-15	2
Difference Treatment	-16	1
Difference Treatment	-11	9
Difference Treatment	-13	4.5
Difference Treatment	-13	4.5
Difference Treatment	-10	11
Difference Control	-11	9
Difference Control	-5	16.5
Difference Control	-4	18
Difference Control	-8	12
Difference Control	-14	3
Difference Control	-5	16.5
Difference Control	-7	13.5
Difference Control	-12	6.5

n	sum of ranks
10	76	Difference Treatment
8	95	Difference Control
18	171	total

95.00	expected value
11.21	standard deviation
-1.65	z, corrected for ties
.9506	p-value (one-tailed, upper)

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P>0.05\Rightarrow \text{Fail to Reject }{H}_0$

Thus, we have no sufficient evidence to conclude that the treatment group show significantly greater improvement then the control group.