Question

The respiratory disturbance index (RDI), which is the number of apneic and hypopneic episodes per hour of sleep, was used as a measure of severity of sleep apnea. An RDI of 5 or more indicates sleep-related breathing disturbances. The RDIs for the females and males samples in a study in US are as follows.

Female				Male
0.1	0.5	0.3	2.3	2.6	19.3	1.4	1.0	0.0	39.2	4.1
2.0	1.4	0.0		0.0	2.1	1.1	5.6	5.0	7.0	2.3
				4.3	7.5	16.5	7.8	3.3	8.9	7.3

At the 1% significance level, do the data provide sufficient evidence to conclude that the female patients are less suffer than male from the apnea disease?

Answer 1

Given
XF bar	0.942857143	XM bar	6.966667
SF	0.946673374	SM	8.899232
nF	7	nM	21

Significance level = 0.01, df = 26

Hypothesis:

H0: μF​ = μM

Ha: μF​ < μM(Left tail)

Critical value:

t critical = -1.7056 (Use t table)

If t stat < t critical, Reject H0

Test:

Sp^2 = ((n1-1)S1^2+(n2-1)S2^2)/(n1+n2-2) = ((7-1)0.9467^2+(21-1)8.8992^2)/(7+21-2) = 61.1271

t stat = (X1 bar-X2 bar )/SQRT(Sp^2*(1/n1 + 1/n2)) = (0.9429-6.9667)/SQRT(61.1271*(1/7 + 1/21)) = -1.7654

Decision:

t stat < t critical, Reject H0

Conclusion:

P value = 0.0026 < 0.01

Reject H0, At the 1% significance level, the data provide sufficient evidence to conclude that the female patients are less suffer than male from the apnea disease