Ina study designed to test the effectiveness 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 the given magnet treatments, the 35 patients had pain scores with a mean of 10.0 and a standard deviation of 2.3.
After being given the sham treatments, the 35 patients had pain scores with a mean of 11.8 and a standard deviation of 2.5. What is the confidence interval
estimate of the population mean u? Round to one decimal place as needed. b) Construct the 90% confidence interval estimate of the mean pain score for patients given the sham treatment. What is the confidence interval estimate of the population mean
u? c) compare the results. Does the treatment with magnets appear to be effective.
t = 1.697
E = 1.697 *2.3 /√35
E = 0.6597
x -E <μ < x bar + E
10- 0.66< μ < 10 + 0.66
9.34 <μ < 10.66
b.
t = 1.697
E = 1.697* 2.5/√35
E = .7171
x bar-E <μ< x bar+E
11.8 -0.72 <μ < 11.8+ 0.72
11.09 < μ < 12.52
c. Since the confidence intervals not overlap, it appears that the magnet treatments are more effective than the sham treatments.
Assume that a test is given to a large number of people but we do not yet know their scores or the shape of the score distribution. Can we be sure that the sampling distribution of the mean for this test will be normally distributed? Why or why not?
Answer this Questi
To calculate the confidence interval estimate of the population mean, we can use the following formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
a) For the treatment with magnets:
Mean (x̄) = 10.0
Standard Deviation (s) = 2.3
Sample Size (n) = 35
To calculate the standard error:
Standard Error = Standard Deviation / sqrt(Sample Size)
Standard Error = 2.3 / sqrt(35) ≈ 0.39
To find the critical value for a 95% confidence interval (since the level of confidence is not given):
Critical Value = 1.96 (for a normally distributed population)
Confidence Interval = 10.0 ± (1.96 * 0.39) ≈ 10.0 ± 0.76
So, the confidence interval estimate of the population mean for the treatment with magnets is approximately 9.24 to 10.76 (rounded to one decimal place).
b) For the sham treatment:
Mean (x̄) = 11.8
Standard Deviation (s) = 2.5
Sample Size (n) = 35
Standard Error = 2.5 / sqrt(35) ≈ 0.42
Confidence Interval = 11.8 ± (1.96 * 0.42) ≈ 11.8 ± 0.83
The 90% confidence interval estimate of the mean pain score for patients given the sham treatment is approximately 10.97 to 12.63 (rounded to one decimal place).
c) By comparing the results, we can determine whether the treatment with magnets appears to be effective. If the confidence interval for the treatment with magnets does not include the mean pain score for the sham treatment, it suggests that the treatment with magnets may be effective. In this case, the confidence interval for the treatment with magnets (9.24 to 10.76) does not overlap with the confidence interval for the sham treatment (10.97 to 12.63). Therefore, based on these results, the treatment with magnets appears to be effective in reducing back pain compared to the sham treatment.
To calculate the confidence interval estimate of the population mean (μ), we can use the formula:
Confidence Interval = (Sample Mean) ± (Critical Value) * (Standard Deviation / √(Sample Size))
a) For the treatment with magnets:
The sample mean is 10.0, the standard deviation is 2.3, and the sample size is 35.
Now, we need to determine the critical value for the desired confidence level. Assuming we want a 95% confidence level, we subtract 1 - 0.95 = 0.05 from 1 and divide it by 2. Checking a z-table or using a calculator, the critical value for a 95% confidence level is approximately 1.96.
Plugging the values into the formula, we get:
Confidence Interval = 10.0 ± (1.96) * (2.3 / √(35))
= 10.0 ± 0.748 (rounded to one decimal place)
So, the confidence interval estimate of the population mean for patients given the magnet treatment is (9.3, 10.7).
b) For the sham treatment:
The sample mean is 11.8, the standard deviation is 2.5, and the sample size is still 35.
Using the same process as above, for a 90% confidence level, the critical value is approximately 1.645.
Plugging the values into the formula, we get:
Confidence Interval = 11.8 ± (1.645) * (2.5 / √(35))
= 11.8 ± 0.853 (rounded to one decimal place)
So, the confidence interval estimate of the population mean for patients given the sham treatment is (10.9, 12.7).
c) To compare the results and determine if the treatment with magnets appears to be effective, we need to check if the confidence intervals overlap.
The confidence interval for the magnet treatment is (9.3, 10.7), while the confidence interval for the sham treatment is (10.9, 12.7).
Since the confidence intervals do not overlap, it suggests that there may be a significant difference between the two treatments. However, determining effectiveness requires additional statistical analysis, such as performing hypothesis tests.