To calculate the upper limit of a 95% confidence interval for the population mean, we use the formula:
\[ CI = \bar{x} \pm z \left(\frac{s}{\sqrt{n}}\right) \]
where:
- \(\bar{x}\) is the sample mean,
- \(z\) is the z-score corresponding to the desired confidence level,
- \(s\) is the sample standard deviation,
- \(n\) is the sample size.
Given:
- Sample mean (\(\bar{x}\)) = 60 years,
- Sample standard deviation (\(s\)) = 8 years,
- Sample size (\(n\)) = 144.
First, we need to find the z-score for a 95% confidence level. For a 95% confidence level, the z-score is approximately 1.96 (you can find this value from z-tables or normal distribution tables).
Now, we can calculate the standard error (SE):
\[ SE = \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{144}} = \frac{8}{12} = \frac{2}{3} \approx 2.667. \]
Next, we calculate the margin of error (ME):
\[ ME = z \times SE = 1.96 \times 2.667 \approx 5.22. \]
Now, we can find the upper limit of the confidence interval:
\[ Upper\ Limit = \bar{x} + ME = 60 + 5.22 \approx 65.22. \]
Thus, the upper limit of the 95% confidence interval for the population mean age of patients who had cataracts removed is approximately 65.22 years.
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