In: Statistics and Probability
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.)
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