Question

A study was conducted to determine whether magnets were effective in treating pain The values represent measurements of pain using the visual analog scale. Assume that both samples are independent simple random samples from populations having normal distributions. Use α= 0.05 significance level to test the claim that those given a sham treatment have pain reductions that vary more than the pain reductions for those treated with magnets.Sham: n=20,¯x=0.44,s=1.24 Magnet: n=20x=0.49,s=0.95a. Identify the test statistic. (Round to two decimal places as needed)b. Use technology to identify the P-value. (Round to three decimal places as needed)c. What is the conclusion for this hypothesis test?A. Reject H0. There is insufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.B. Reject H0. There is sufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.C. Fail to reject H0. There is sufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.D. Fail to reject H0. There is insufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.

Answer 1

a. Test statistic t = -0.14

b. P-value = 0.443

c. D. Fail to reject H0. There is insufficient evidence to support the claim that those given a sham treatment have pain reductions that vary more than those treated with magnets.

Explanation:

This is a hypothesis test for the difference between populations means.

The claim is that those given a sham treatment have pain reductions that vary more than the pain reductions for those treated with magnets.

Then, the null and alternative hypothesis are:

`H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2< 0`

The significance level is α=0.05.

The sample 1 (sham), of size n1=20 has a mean of 0.44 and a standard deviation of 1.24.

The sample 2 (magnet), of size n2=20 has a mean of 0.49 and a standard deviation of 0.95.

The difference between sample means is Md=-0.05.

`M_d=M_1-M_2=0.44-0.49=-0.05`

The estimated standard error of the difference between means is computed using the formula:

`s_(M_d)=\sqrt{(\sigma_1^2+\sigma_2^2)/(n)}=\sqrt{(1.24^2+0.95^2)/(20)}\\\\\\s_(M_d)=\sqrt{(2.4401)/(20)}=√(0.122)=0.3493`

Then, we can calculate the t-statistic as:

`t=(M_d-(\mu_1-\mu_2))/(s_(M_d))=(-0.05-0)/(0.3493)=(-0.05)/(0.3493)=-0.14`

The degrees of freedom for this test are:

`df=n_1+n_2-2=20+20-2=38`

This test is a left-tailed test, with 38 degrees of freedom and t=-0.14, so the P-value for this test is calculated as (using a t-table):

`P-value=P(t<-0.14)=0.443`

As the P-value (0.443) is greater than the significance level (0.05), the effect is not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that those given a sham treatment have pain reductions that vary more than the pain reductions for those treated with magnets.