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/cm2/min) was measured. For m = 8 subjects with the syndrome, the average heat output was x = 0.64, and for n = 8 nonsufferers, the average output was 2.09. Let ?1 and ?2 denote the true average heat outputs for the sufferers and nonsufferers, respectively. Assume that the two distributions of heat output are normal with ?1 = 0.1 and ?2 = 0.4.

Answer 1

The null hypothesis is rejected.

There is enough evidence to support the claim that the average heat output for people with the syndrome is lower than for the non-sufferers. (P-value = 0.00000005).

Explanation:

This is a hypothesis test for the difference between populations means.

The claim is that the average heat output for people with the syndrome is lower than for the non-sufferers.

Then, the null and alternative hypothesis are:

`H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2< 0`

The significance level is asusmed to be α=0.05.

The sample 1 (subjects with syndrome), of size n1=8 has a mean of 0.64 and a standard deviation of 0.1.

The sample 2 (non-sufferers), of size n2=8 has a mean of 2.09 and a standard deviation of 0.4.

The difference between sample means is Md=-1.45.

`M_d=M_1-M_2=0.64-2.09=-1.45`

The estimated standard error of the difference between means is computed using the formula:

`s_(M_d)=\sqrt{(\sigma_1^2+\sigma_2^2)/(n)}=\sqrt{(0.1^2+0.4^2)/(8)}\\\\\\s_(M_d)=\sqrt{(0.17)/(8)}=√(0.021)=0.146`

Then, we can calculate the t-statistic as:

`t=(M_d-(\mu_1-\mu_2))/(s_(M_d))=(-1.45-0)/(0.146)=(-1.45)/(0.146)=-9.9469`

The degrees of freedom for this test are:

`df=n_1+n_2-1=8+8-2=14`

This test is a left-tailed test, with 14 degrees of freedom and t=-9.9469, so the P-value for this test is calculated as (using a t-table):

`P-value=P(t<-9.9469)=0.00000005`

As the P-value (0.00000005) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the average heat output for people with the syndrome is lower than for the non-sufferers.