Persons having Raynaud\'s syndrome are apt to suffer a sudden impairment of bloo

Question

Persons having Raynaud's syndrome are apt to suffer a sudden impairment of blood circulation in fingers and toes. In an experiment to study the extent of this impairment, each subject immersed a forefinger in water and the resulting heat output (cal/cm^2/min) was measured. For m = 8 subjects with the syndrome, the average heat output was x = 0.61, and for n = 8 no sufferers, the average output was 2.06. Let mu_1 and mu_2 denote the true average heat outputs for the sufferers and no sufferers, respectively. Assume that the two distributions of heat output are normal with sigma_1 = 0.2 and sigma_2 = 0.4. Consider testing H_0: = -1*0 versus H_a: mu_1 - mu_2

Explanation / Answer

H0: 1 – 2 = -1

H1: 1 – 2 < -1

            

Assuming population variances are equal, we would have to calculate pooled-variance t-Test

Sp^2= (n1-1)S1^2+(n2-1)S2^2/(n1-1)+(n2-1)

         = (8-1)*0.2^2+(8-1)*0.4^2/7+7

         =0.1

tSTAT=(X1-X2)-(µ1-µ2)/Sp^2(1/n1+1/n2)

       =(0.61-2.06)-(-1)/0.1(1/8+1/8)

       =-0.45/0.1581

       =-2.846

tSTAT=(X1-X2)-(µ1-µ2)/Sp^2(1/n1+1/n2)

       =(0.61-2.06)-(-1.5)/0.1(1/8+1/8)

       =0.05/0.1581

       =0.3163

Looking up the table, we get the value of 0.6255 and that is the probability of type 2 error.

1.645^2*-1.5^2/0.1=61 is the required sample size

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