Question

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 0.050.05 significance level for both parts. Treatment Sham muμ mu 1μ1 mu 2μ2 n 1212 1212 x overbarx 0.480.48 0.440.44 s 0.830.83 1.381.38 a. Test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment. What are the null and alternative​ hypotheses? A. Upper H 0H0​: mu 1μ1less thanmu 2μ2 The test​ statistic, t, is nothing. ​(Round to two decimal places as​ needed.) The​ P-value is nothing. ​(Round to three decimal places as​ needed.) State the conclusion for the test. ▼ Reject Fail to reject the null hypothesis. There ▼ is not is 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 standard deviation sample mean for those treated with magnets is ▼ equal to less than greater than the sample mean for those given a sham​ treatment, it ▼ is is not 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. nothingless than

Answer 1

Solution:-

a)

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

Null hypothesis: u_Treatment< u_Placebo
Alternative hypothesis: u_Treatment > u_Placebo

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. 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/n₁) + (s₂²/n₂)]
SE = 0.4649
DF = 22

t = [ (x₁ - x₂) - d ] / SE

t = 0.09

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ 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.

The observed difference in sample means produced a t statistic of 0.09.

Therefore, the P-value in this analysis is 0.465

Interpret results. Since the P-value (0.465) is greater than the significance level (0.05), we failed to 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 greater than the sample mean for those given a sham​ treatment, it is valid to argue that magnets might appear to be effective if the sample sizes are larger.

b) 95% confidence interval for the difference in the population mean is C.I = ( - 0.924, 1.004).

C.I = (0.48 - 0.44) + 2.074*0.4649

C.I = 0.04 + 0.96415

C.I = ( - 0.924, 1.004)