To find the 99.9% confidence interval for the population mean \(\mu\), we'll use the following formula for the confidence interval:
\[ \text{Confidence Interval} = \bar{x} \pm z^* \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \(\bar{x} = 85.8\) (the sample mean),
- \(s = 13.3\) (the sample standard deviation),
- \(n = 43\) (the sample size),
- \(z^*\) is the critical value for 99.9% confidence level.
Step 1: Find the critical value \(z^*\)
For a 99.9% confidence interval, the two-tailed alpha level \(\alpha\) is:
\[ \alpha = 1 - 0.999 = 0.001 \]
Thus, the area in each tail is:
\[ \frac{\alpha}{2} = \frac{0.001}{2} = 0.0005 \]
Using the standard normal distribution table or a calculator, we find the critical z-value corresponding to a cumulative probability of \(1 - 0.0005 = 0.9995\). The critical value \(z^*\) is approximately:
\[ z^* \approx 3.291 \]
Step 2: Calculate the margin of error
Next, we calculate the standard error (SE) and then the margin of error (ME):
\[ \text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{13.3}{\sqrt{43}} \approx \frac{13.3}{6.557} \approx 2.03 \]
Now calculating the margin of error:
\[ \text{Margin of Error} = z^* \cdot \text{Standard Error} = 3.291 \cdot 2.03 \approx 6.68 \]
Step 3: Calculate the confidence interval
Finally, we compute the confidence interval:
\[ \text{Confidence Interval} = \bar{x} \pm \text{Margin of Error} \]
Calculating the lower and upper limits:
\[ \text{Lower Limit} = 85.8 - 6.68 \approx 79.12 \]
\[ \text{Upper Limit} = 85.8 + 6.68 \approx 92.48 \]
Conclusion
Thus, the 99.9% confidence interval for the population mean \(\mu\) is:
\[ \text{99.9% C.I.} = (79.1, 92.5) \]
Therefore, the final answer is:
\[ \boxed{(79.1, 92.5)} \]
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