Question

In a study designed to test the effectiveness of magnets for treating back​ pain, 40 patients were given a treatment with magnets and also a sham treatment without magnets. Pain was measured using a scale from 0​ (no pain) to 100​ (extreme pain). After given the magnet​ treatments, the 40 patients had pain scores with a mean of 12.0 and a standard deviation of 2.4. After being given the sham​ treatments, the 40 patients had pain scores with a mean of 12.4 and a standard deviation of 2.6.

a. Construct the90% confidence interval estimate of the mean pain score for patients given the magnet treatment.

What is the confidence interval estimate of the population mean μ​?

b. Construct the 90% confidence interval estimate of the mean pain score for patients given the sham treatment.

What is the confidence interval estimate of the population mean μ​?

Compare the results. Does the treatment with magnets appear to be​ effective?

A.Since the confidence intervals overlap, it appears that the magnet treatments are no more effective than the sham treatments.

B.Since the confidence intervals do not nbsp do not ​overlap, it appears that the magnet treatments are no more effective than the sham treatments.

C.Since the confidence intervals ​overlap, it appears that the magnet treatments are less effective than the sham treatments

D.Since the confidence intervals do not nbsp do not ​overlap, it appears that the magnet treatments are less effective than the sham treatments.

Answer 1

a)

sample mean, xbar = 12
sample standard deviation, s = 2.4
sample size, n = 40
degrees of freedom, df = n - 1 = 39

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, tc = t(α/2, df) = 1.68

ME = tc * s/sqrt(n)
ME = 1.68 * 2.4/sqrt(40)
ME = 0.638

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (12 - 1.68 * 2.4/sqrt(40) , 12 + 1.68 * 2.4/sqrt(40))
CI = (11.36 , 12.64)

b)

sample mean, xbar = 12.4
sample standard deviation, s = 2.6
sample size, n = 40
degrees of freedom, df = n - 1 = 39

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, tc = t(α/2, df) = 1.685

ME = tc * s/sqrt(n)
ME = 1.685 * 2.6/sqrt(40)
ME = 0.693

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (12.4 - 1.685 * 2.6/sqrt(40) , 12.4 + 1.685 * 2.6/sqrt(40))
CI = (11.71 , 13.09)

C.Since the confidence intervals ​overlap, it appears that the magnet treatments are less effective than the sham treatments