QUESTION IMAGE
Question
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).
$-0.77 < \mu_1 - \mu_2 < 0.93$
(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 effective in treating back pain, because the confidence interval contains only positive values.
b. it appears that magnets are not effective in treating back pain, because 0 is in the confidence interval.
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 not effective in treating back pain, because the p - value is less than the significance level.
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. no, because magnets already appear to be effective.
d. yes, because increasing the sample size will decrease the p - value.
For part c (Does it appear that magnets are effective...): The confidence interval for \(\mu_1 - \mu_2\) is \(-0.77 < \mu_1 - \mu_2 < 0.93\). Since 0 is within this interval, we can't conclude a significant difference (magnets effective). So option B: "It appears that magnets are not effective in treating back pain, because 0 is in the confidence interval" is correct.
For the "Is it valid to argue..." part: Larger sample sizes increase statistical power (ability to detect effects). If the initial sample didn't show significance, larger samples might reveal an effect (since small samples can miss true effects). So the valid argument is that magnets might appear effective with larger samples. Among options A - D, the reasoning about sample size and P - value (A says increasing sample size increases P - value, which is wrong; B says it increases effectiveness, which is about the effect, not directly; C says magnets already appear effective, which they don't from the interval; D says increasing sample size decreases P - value, which is correct for detecting effects, but the "valid to argue" is about whether larger samples could show effectiveness. Wait, re - reading: The question is "Is it valid to argue that magnets might appear to be effective if the sample sizes are larger?". Larger sample sizes can detect smaller effects (lower P - value when there is a true effect). So the correct answer is B: "Yes, because increasing the sample size will increase the effectiveness" (more accurately, increase the ability to detect effectiveness). Wait, no—sample size increases statistical power, so if there is a true effect, larger samples are more likely to detect it (lower P - value). So the valid argument is yes, because larger samples can show effectiveness (by reducing P - value to be significant). Wait, the options:
A. No, because increasing sample size increases P - value (wrong, larger n decreases standard error, so test statistic increases, P - value decreases if there is an effect).
B. Yes, because increasing sample size will increase effectiveness (sample size doesn't increase the actual effectiveness, but the ability to detect it).
C. No, because magnets already appear effective (they don't, since 0 is in the interval).
D. Yes, because increasing sample size decreases P - value (this is about the statistical test, and if there is a true effect, lower P - value with larger n can show significance). But the question is about "argue that magnets might appear to be effective". So the valid argument is yes, because larger samples can detect the effect (by decreasing P - value to be significant, making them appear effective). So option B? Wait, no—option B says "increase the effectiveness", which is incorrect. Option D says "Yes, because increasing the sample size will decrease the P - value". If the P - value decreases (with larger n) to be below the significance level, then we would reject the null and conclude effectiveness. So the valid argument is yes, because larger sample size can decrease P - value, leading to the appearance of effectiveness. So option D? Wait, the options:
A. No, because increasing sample size increases P - value (wrong).
B. Yes, because increasing sample size will increase effectiveness (sample size doesn't change the actual effectiveness of the treatment, just the ability to detect it).
C. No, because magnets already appear effective (they don't).
D. Yes, because increasing sample size will decrease the P - value (correct, because larger n gives more precise estimates, so if there is a true effect, the P -…
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For the "Does it appear that magnets are effective..." question: B. It appears that magnets are not effective in treating back pain, because 0 is in the confidence interval.
For the "Is it valid to argue..." question: D. Yes, because increasing the sample size will decrease the P - value.