Persons having Raynaud\'s syndrome are apt to suffer a sudden impairment of bloo
ID: 3232915 • Letter: P
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 = 10 subjects with the syndrome, the average heat output was x bar = 0.61 and for n = 10 nonsufferers, the average output was 2.04. Let mu_1 and mu_2 denote the true average heat outputs for the sufferers and nonsufferers 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: mu_1 - mu_2 = -1.0 versus H_a: mu_1 - mu_2Explanation / Answer
Solution:-
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = - 1.0
Alternative hypothesis: 1 - 2 < - 1.0
Note that these hypotheses constitute a one-tailed test.
Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
S.E = sqrt[(s12/n1) + (s22/n2)]
S.E = 0.1414
DF = 18
t = [ (x1 - x2) - d ] / SE
t = - 3.04
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.
The observed difference in sample means (10) produced a t statistic of - 3.04. We use the t Distribution Calculator to find P(t > - 3.05) = 0.0035
Therefore, the P-value in this analysis is 0.0035
Interpret results. Since the P-value (0.0035) is less than the significance level (0.01), we have to reject the null hypothesis.
Reject H0. The data suggests that the average heat output for sufferers is more than 1 cal/cm2/min below that of non sufferers.
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