Question

The table below shows the number of hours per day 11 patients suffered from headaches before and after 7 weeks of soft tissue therapy. At α =​0.01, is there enough evidence to conclude that soft tissue therapy helps to reduce the length of time patients suffer from​ headaches? Assume the samples are random and​ dependent, and the population is normally distributed. USE THE CRITICAL REGION METHOD

Patient: 1 2 3 4 5 6 7 8 9 10 11

Daily Headache HRS.( Before) 3.1 , 2.6 , 3.6 , 3.9, 1.8 , 3.3 , 3.8, 1.8 , 2.5, 3.9 , 1.6

Daily Headache HRS.(After) 1.8 , 1.8, 1.4 , 2.2, 1.2 , 2.4 ,3.0 ,1.3 , 1.6 , 1.3, 1.1

Answer 1

The table given below ,

Patient	Before (X)	After(Y)	di=X-Y	di^2
1	3.1	1.8	1.3	1.69
2	2.6	1.8	0.8	0.64
3	3.6	1.4	2.2	4.84
4	3.9	2.2	1.7	2.89
5	1.8	1.2	0.6	0.36
6	3.3	2.4	0.9	0.81
7	3.8	3	0.8	0.64
8	1.8	1.3	0.5	0.25
9	2.5	1.6	0.9	0.81
10	3.9	1.3	2.6	6.76
11	1.6	1.1	0.5	0.25
		Sum	12.8	19.94

From table ,

Let ,

The null and alternative hypothesis is ,

The test is one-tailed test.

Use the paired t-test.

Now , df=degrees of freedom=n-1=11-1=10

The critical value is ,

; From t-table

The test statistic is ,

Decision : Here , the value of the test statistic lies in the rejection region.

Therefore , reject Ho.

Conclusion : There is enough evidence to conclude that soft tissue therapy helps to reduce the length of time patients suffer from headaches.