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:
\[ (1.569, 2.031) \]