Question

In a study designed to test the effectiveness of acupuncture for treating migraine, 142 subjects were treated with acupuncture. The numbers of migraine attacks for the treatment group had a mean of 1.8 and a standard deviation of 1.4. Construct a 95% confidence interval estimate of the mean number of migraine attacks for all people treated with acupuncture.

Answer 1

To construct a 95% confidence interval for the mean number of migraine attacks for all people treated with acupuncture, we can use the formula for the confidence interval of the mean:

\[
\text{Confidence Interval} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)
\]

Where:
- \(\bar{x}\) is the sample mean
- \(z\) is the z-value corresponding to the desired confidence level (for 95%, \(z \approx 1.96\))
- \(s\) is the sample standard deviation
- \(n\) is the sample size

Given:
- \(\bar{x} = 1.8\)
- \(s = 1.4\)
- \(n = 142\)

First, we need to calculate the standard error (SE):

\[
SE = \frac{s}{\sqrt{n}} = \frac{1.4}{\sqrt{142}} \approx \frac{1.4}{11.83} \approx 0.118
\]

Next, we calculate the margin of error (ME):

\[
ME = z \cdot SE \approx 1.96 \cdot 0.118 \approx 0.231
\]

Now, we can find the confidence interval:

\[
\text{Confidence Interval} = 1.8 \pm 0.231
\]

This gives us:

\[
\text{Lower Limit} = 1.8 - 0.231 \approx 1.569
\]
\[
\text{Upper Limit} = 1.8 + 0.231 \approx 2.031
\]

Thus, the 95% confidence interval estimate of the mean number of migraine attacks for all people treated with acupuncture is approximately:

\[
(1.569, 2.031)
\]