Question

the p - value is 0.466. (round to three decimal places as needed.)
state the conclusion for the test.
b. 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.
b. construct a confidence interval appropriate for the hypothesis test in part (a).

• 0.76 < μ₁ - μ₂ < 0.84 (round to two decimal places as needed.)

c. does it appear that magnets are effective in treating back pain? is it valid to argue that magnets might appear to be effective if the sample sizes are larger? choose the correct answer below.
a. it appears that magnets are not effective in treating back pain, because 0 is in the confidence interval.
b. it appears that magnets are not effective in treating back pain, because the p - value is less than the significance level.
c. it appears that magnets are effective in treating back pain, because the p - value is greater than the significance level.
d. it appears that magnets are effective in treating back pain, because the confidence interval contains only positive values.
is it valid to argue that magnets might appear to be effective if the sample sizes are larger? choose the correct answer below.
a. no, because increasing the sample size will increase the p - value.
b. yes, because increasing the sample size will increase the effectiveness.
c. yes, because increasing the sample size will decrease the p - value.
d. no, because the magnets already appear to be effective.

Explanation:

Part b (Confidence Interval and Effectiveness of Magnets)

To determine if magnets appear effective, we check the confidence interval for \(\mu_1 - \mu_2\) (where \(\mu_1\) is the mean pain reduction for magnet treatment and \(\mu_2\) for sham treatment). The interval is \(-0.76 < \mu_1 - \mu_2 < 0.84\). A confidence interval containing 0 means we can't conclude a significant difference (since 0 is within the interval, the difference could be zero). So magnets do not appear effective. Option A says "It appears that magnets are not effective in treating back pain, because 0 is in the confidence interval"—this matches the reasoning.

Larger sample sizes increase statistical power (ability to detect effects). When sample size increases, the standard error decreases, so the P - value (which is related to the test statistic) will decrease (since the test statistic \(t=\frac{\bar{x}_1 - \bar{x}_2}{SE}\) will increase in magnitude, leading to a smaller P - value). A smaller P - value makes it more likely to reject the null hypothesis (if there is a true effect). So it is valid to argue magnets might be effective with larger samples because increasing sample size decreases the P - value (Option C: "Yes, because increasing the sample size will decrease the P - value").

Answer:

A. It appears that magnets are not effective in treating back pain, because 0 is in the confidence interval.

Part c (Validity of Arguing with Larger Sample Sizes)