Question

test was conducted to determine the effectiveness of using an anti-inflammatory cream on delayed-onset muscle soreness. A random sample of ten patients was treated with the cream on one arm and with a placebo on the other (control) arm. After four days, a measure of muscle soreness was then taken for each patient on each arm. The results are: Patient 1 2 3 4 5 6 7 8 9 10 Control Arm 46 22 10 14 26 29 29 47 20 13 Treated Arm 2 32 30 3 14 32 2 39 18 2 At α=0.01 level of significance, would you say there is less soreness in the treated arm? Your discussion must include null and alternative hypotheses, check of assumptions (with appropriate plots), the value of the test statistic, the P-value, and your conclusion in the context of the data.

Answer 1

Solution:-

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

Null hypothesis: u_d< 0

Alternative hypothesis: u_d > 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 matched-pairs t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

s = sqrt [ (\sum (d_i - d)² / (n - 1) ]

s = 17.239

SE = s / sqrt(n)

S.E = 5.452

DF = n - 1 = 10 -1

D.F = 9

t = [ (x₁ - x₂) - D ] / SE

t = 1.82

where d_i is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

Since we have a one-tailed test, the P-value is the probability that a t statistic having 9 degrees of freedom is greater than 1.816

Thus, the P-value = 0.051

Interpret results. Since the P-value (0.051) is greater than the significance level (0.01), we have to accept the null hypothesis.

Do not reject H0.

From the above test we do not have sufficient evidence in the favor of the claim that there is less soreness in the treated arm.