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/cm2/min) was measured. For m = 10 subjects with the syndrome, the average heat output was x = 0.64, and for n = 10 nonsufferers, the average output was 2.02. 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 = 0.2 and 2 = 0.3.

(a) Consider testing H0: 1 2 = 1.0 versus Ha: 1 2 < 1.0 at level 0.01. Describe in words what Ha says, and then carry out the test.

Ha says that the average heat output for sufferers is the same as that of non-sufferers. Ha says that the average heat output for sufferers is less than 1 cal/cm2/min below that of non-sufferers.     Ha says that the average heat output for sufferers is more than 1 cal/cm2/min below that of non-sufferers.

Calculate the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

State the conclusion in the problem context.

Reject H0. The data suggests that the average heat output for sufferers is the same as that of non-sufferers. Fail to reject H0. The data suggests that the average heat output for sufferers is the same as that of non-sufferers.     Reject H0. The data suggests that the average heat output for sufferers is more than 1 cal/cm2/min below that of non-sufferers. Fail to reject H0. The data suggests that the average heat output for sufferers is less than 1 cal/cm2/min below that of non-sufferers.

(b) What is the probability of a type II error when the actual difference between 1 and 2 is

1 2 = 1.2?

(Round your answer to four decimal places.)
5

(c) Assuming that m = n, what sample sizes are required to ensure that = 0.1 when

1 2 = 1.2?

(Round your answer up to the nearest whole number.)
6 subjects

Explanation / Answer

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: H0: 1 2 = 1.0
Alternative hypothesis: 1 2 < 1.0

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the mean difference between sample means is too small.

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).

SE = sqrt[(s12/m) + (s22/n)]
SE = sqrt[(0.22/10) + (0.32/10] = 0.114

DF = (s12/m + s22/n)2 / { [ (s12 / m)2 / (m - 1) ] + [ (s22 / n)2 / (n - 1) ] }
DF = (0.22/10 + 0.32/10)2 / { [ (0.22 / 10)2 / (9) ] + [ (0.32 / 10)2 / (9) ] }
DF = 0.000169 / (0.00000178 + 0.000009) = 15.68

t = [ (x1 - x2) - d ] / SE = [(0.64 - 2.02) - (-1)] / 0.114 = -3.33

where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, m is the size of sample 1, n 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.

Here is the logic of the analysis: Given the alternative hypothesis (1 - 2 < -1), we want to know whether the observed difference in sample means is small enough (i.e., sufficiently less than -1) to cause us to reject the null hypothesis.

We use the t Distribution Calculator to find P(t < -3.33)

The P-Value is 0.002171.

Interpret results. Since the P-value (0.002171) is smaller than the significance level (0.01), we can reject the null hypothesis. Thus, the average heat output for sufferers is less than 1 cal/cm2/min below that of non-sufferers.

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