Question

4. Researchers conducted a study to determine whether magnets are effective in treating back pain. The results are shown in the table for the treatment​ (with magnets) group and the sham​ (or placebo) group. The results are a measure of reduction in back pain. Assume that the two samples are independent simple random samples selected from normally distributed​ populations, and do not assume that the population standard deviations are equal. Complete parts​ (a) and​ (b) below. Use a 0.05 significance level for both parts.

	Treatment	Sham
µ	µ1	µ2
N	11	11
xˉ	0.52	0.43
S	0.76	1.15

Test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.

a.H0: µ1=µ2

H1: µ1≠µ2

b. H0: µ1<µ2

H1: µ1≥µ2

c. H0: µ1=µ2

H1: µ1>µ2

d. H0: µ1≠µ2

H1: µ1<µ2

What are the null and alternative​ hypotheses?

The test​ statistic, t, is ____

​(Round to two decimal places as​ needed.)

The​ P-value is ____

​(Round to three decimal places as​ needed.)

State the conclusion for the test.

(Fail to reject/reject) the null hypothesis. There (is/is not) sufficient evidence to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.

Is it valid to argue that magnets might appear to be effective if the sample sizes are​ larger?

Since the (sample mean/sample standard deviation) for those treated with magnets is (greater than/less than/equal to) the sample mean for those given a sham​ treatment, it (is not/is) valid to argue that magnets might appear to be effective if the sample sizes are larger.

b. Construct a confidence interval suitable for testing the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.

___<µ1-µ2<___

​(Round to three decimal places as​ needed.)

Answer 1

Option - C) H₀:

                  H₁:

The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                            = (0.52 - 0.43)/sqrt((0.76)^2/11 + (1.15)^2/11)

                            = 0.22

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

    = ((0.76)^2/11 + (1.15)^2/11)^2/(((0.76)^2/11)^2/10 + ((1.15)^2/11)^2/10)

    = 17

P-value = P(T > 0.22)

             = 1 - P(T < 0.22)

             = 1 - 0.5858

             = 0.4142 = 0.414

Since the P-value is greater than the significance level (0.4142 > 0.05), so we should not reject the null hypothesis.

Fail to reject the null hypothesis. There is not sufficient evidence to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.

Since the sample mean for those treated with magnets is equal to the sample mean for those given a sham treatment , it is not valid to argue that magnets might appear to be effective if the sample sizes are larger.

b) At 95% confidence interval the critical value is t^* = 2.110

The 95% confidence interval for is

() +/- t^* * sqrt(s1^2/n1 + s2^2/n2)

= (0.52 - 0.43) +/- 2.110 * sqrt((0.76)^2/11 + (1.15)^2/11)

= 0.09 +/- 0.877

= -0.787, 0.967