In a study designed to test the effectiveness of magnets for treating back pain, 35 patients were given a treatment with magnets and also a sham treatment without magnets. Pain was measured using a scale from 0 (no pain) to 100 (extreme pain). After given the magnet treatments, the 35 patients had pain scores with a mean of 9.0 and a standard deviation of 2.2. After being given the sham treatments, the 35 patients had pain scores with a mean of 9.2 and a standard deviation of 2.5. Complete parts (a) through (c) below.
a. Construct the 95% confidence interval estimate of the mean pain score for patients given the magnet treatment.
What is the confidence interval estimate of the population mean mu?
(Round to one decimal place as needed.)
b. Construct the 95% confidence interval estimate of the mean pain score for patients given the sham treatment.
What is the confidence interval estimate of the population mean mu?
(Round to one decimal place as needed.)
c. Compare the results. Does the treatment with magnets appear to be effective?
A.
Since the confidence intervals overlap, it appears that the magnet treatments are less effective than the sham treatments.
B.
Since the confidence intervals overlap, it appears that the magnet treatments are no more effective than the sham treatments.
C.
Since the confidence intervals do not overlap, it appears that the magnet treatments are less effective than the sham treatments.
D.
Since the confidence intervals do not overlap, it appears that the magnet treatments are no more effective than the sham treatments.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
However, I will give you a start.
95% = mean ± 1.96SD
To construct the confidence intervals and compare the results, we can follow these steps:
a. Constructing the confidence interval estimate of the mean pain score for patients given the magnet treatment:
To construct the confidence interval, we need the sample mean (x̄), the standard deviation (s), the sample size (n), and the confidence level.
Given in the question:
Sample mean (x̄) = 9.0
Standard deviation (s) = 2.2
Sample size (n) = 35
Confidence level = 95%
The formula for calculating the confidence interval for the population mean is:
Confidence Interval = x̄ ± (Zscore * (s/√n))
To find the Zscore corresponding to a 95% confidence level, we need to use a Z-table or a calculator. The Zscore corresponding to a 95% confidence level is approximately 1.96.
Plugging in the values:
Confidence Interval = 9.0 ± (1.96 * (2.2/√35))
Calculating this, we get:
Confidence Interval = (8.3, 9.7)
Therefore, the 95% confidence interval estimate of the mean pain score for patients given the magnet treatment is (8.3, 9.7).
b. Constructing the confidence interval estimate of the mean pain score for patients given the sham treatment:
Using the same steps and the information from the question:
Sample mean (x̄) = 9.2
Standard deviation (s) = 2.5
Sample size (n) = 35
Confidence level = 95%
Plugging in the values:
Confidence Interval = 9.2 ± (1.96 * (2.5/√35))
Calculating this, we get:
Confidence Interval = (8.4, 10.0)
Therefore, the 95% confidence interval estimate of the mean pain score for patients given the sham treatment is (8.4, 10.0).
c. Comparing the results to determine the effectiveness of the magnet treatment:
To compare the results, we need to examine the confidence intervals. If the confidence intervals overlap, it suggests that there is no significant difference between the two treatments. If the confidence intervals do not overlap, it suggests that there is a significant difference between the two treatments.
Looking at the confidence intervals:
- The confidence interval for magnet treatment is (8.3, 9.7)
- The confidence interval for sham treatment is (8.4, 10.0)
Since the confidence intervals overlap, it indicates that the magnet treatments are no more effective than the sham treatments. Therefore, the correct answer is B. Since the confidence intervals overlap, it appears that the magnet treatments are no more effective than the sham treatments.