In a study designed to test the effectiveness of magnets for treating back pain, 40 patients were given a treatment with magnets and also a sham treatment without magnets. Pain was measured using a scale from 0 to 100. After given the magnet treatments, the 40 patients had pain scores with a mean of 9.0 and a standard deviation of 2.3. After being given the sham treatment, the 40 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 99% confidence interval estimate of the mean pain score for patients given the magnet treatment. What is the confidence interval of the population mean µ?
?<µ<? Round to one decimal place as needed
b. Construct the 99% confidence interval estimate of the mean pain score for the patients given the sham treatment. What is the confidence interval estimate of the population mean µ?
?<µ<? Round to one decimal place as needed
c. Compare the results. Does the treatment with magnets appear to be effective?
t = 2.75
E = 2.75*2.3/ √40
E = 1.00
xbar -E <μ < x bar + E
9 - < μ < 9 + 1.00
8.00 μ < 10.00
b.
t = 2.75
E = 2.75 * 2.5/√40
E = 1.089
x bar -E<μ x bar +E
9.2- 1.089< μ < 9.2 +1.089
8.111< μ < 10.287
c. Since the confidence intervals overlap, it appears that the magnet treatments are no more effective than the sham treatments.
To construct confidence intervals for the mean pain score, we need to use the sample data provided, along with the formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √(Sample Size))
Let's calculate each confidence interval step-by-step:
a. Confidence Interval for the Mean Pain Score with Magnet Treatment:
Sample Mean = 9.0 (mean of pain scores)
Standard Deviation = 2.3 (standard deviation of pain scores)
Sample Size = 40
First, we need to find the critical value corresponding to the desired confidence level. Since we want a 99% confidence interval, we need to find the z-score which leaves a 0.5% margin on each tail of the normal distribution. By looking up this value in a standard normal distribution table or using statistical software, we find the critical value to be approximately 2.576.
Now, we can calculate the confidence interval:
Confidence Interval = 9.0 ± (2.576) * (2.3 / √(40))
Confidence Interval ≈ 9.0 ± 1.864
Confidence Interval ≈ 7.136 to 10.864
Therefore, the 99% confidence interval estimate for the mean pain score of patients given the magnet treatment is 7.136 to 10.864.
b. Confidence Interval for the Mean Pain Score with Sham Treatment:
Sample Mean = 9.2 (mean of pain scores)
Standard Deviation = 2.5 (standard deviation of pain scores)
Sample Size = 40
Using the same critical value of 2.576, we can calculate the confidence interval:
Confidence Interval = 9.2 ± (2.576) * (2.5 / √(40))
Confidence Interval ≈ 9.2 ± 1.982
Confidence Interval ≈ 7.218 to 11.182
Thus, the 99% confidence interval estimate for the mean pain score of patients given the sham treatment is 7.218 to 11.182.
c. Comparing the Results:
To determine if the magnet treatment appears to be effective, we can compare the confidence intervals. If the confidence intervals overlap significantly or contain similar values, it suggests that there is no significant difference between the two treatment groups.
Here, we observe that the confidence intervals for both groups overlap substantially. The magnet treatment group has a confidence interval of 7.136 to 10.864, while the sham treatment group has a confidence interval of 7.218 to 11.182. This indicates that there is no strong evidence to suggest a significant difference in pain reduction between the two treatments.
In conclusion, based on the comparison of confidence intervals, the treatment with magnets does not appear to be effective for reducing back pain compared to the sham treatment.
To construct confidence intervals for the mean pain scores with magnet treatment and sham treatment, we can use the formula:
Confidence Interval = Mean ± (Critical Value * Standard Error)
where the critical value depends on the desired confidence level and the standard error is calculated as the standard deviation divided by the square root of the sample size.
Given the information in the question:
For magnet treatment:
Sample mean (x̄) = 9.0
Standard deviation (s) = 2.3
Sample size (n) = 40
For sham treatment:
Sample mean (x̄) = 9.2
Standard deviation (s) = 2.5
Sample size (n) = 40
a. To construct the 99% confidence interval estimate of the mean pain score for patients given the magnet treatment, we need to find the critical value for a 99% confidence level. This can be looked up in a standard normal distribution table or calculated using a statistical software. For a 99% confidence level, the critical value is approximately 2.626.
Standard error (SE) = s / sqrt(n) = 2.3 / sqrt(40) = 0.3636 (approx.)
Confidence Interval = 9.0 ± (2.626 * 0.3636)
= 9.0 ± 0.9548
= [8.0452, 9.9548]
Therefore, the 99% confidence interval of the mean pain score for patients given the magnet treatment is [8.0, 9.9] (rounded to one decimal place).
b. To construct the 99% confidence interval estimate of the mean pain score for the patients given the sham treatment, we use the same method:
Standard error (SE) = s / sqrt(n) = 2.5 / sqrt(40) = 0.3953 (approx.)
Confidence Interval = 9.2 ± (2.626 * 0.3953)
= 9.2 ± 1.0376
= [8.1624, 10.2376]
Therefore, the 99% confidence interval of the mean pain score for patients given the sham treatment is [8.2, 10.2] (rounded to one decimal place).
c. To compare the results, we can look at the overlap between the two confidence intervals. Since the confidence intervals for both treatment groups overlap, it suggests that there is no significant difference in the effectiveness of magnet treatment and sham treatment for back pain. In other words, the treatment with magnets does not appear to be more effective than the sham treatment in reducing back pain based on the given data.