To calculate the 99% confidence interval for the mean length of the sheet insulation, we can use the following formula for a confidence interval for the population mean when the population standard deviation is known:
\[ CI = \bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right) \]
where:
- \(\bar{x}\) is the sample mean,
- \(z\) is the z-value corresponding to the confidence level,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
Given that:
- Sample mean (\(\bar{x}\)) = 15.14 feet
- Population standard deviation (\(\sigma\)) = 0.2 feet
- Sample size (\(n\)) = 60
For a 99% confidence level, we look up the z-value that corresponds to 0.005 in each tail (since \(1 - 0.99 = 0.01\)), which is approximately:
\[ z \approx 2.576 \]
Now, we first calculate the standard error of the mean (SEM):
\[ SEM = \frac{\sigma}{\sqrt{n}} = \frac{0.2}{\sqrt{60}} \approx \frac{0.2}{7.745966} \approx 0.0258 \]
Next, we calculate the margin of error (ME):
\[ ME = z \times SEM = 2.576 \times 0.0258 \approx 0.0666 \]
Now we can construct the confidence interval:
\[ CI = \bar{x} \pm ME = 15.14 \pm 0.0666 \]
Calculating the lower and upper bounds:
- Lower bound = 15.14 - 0.0666 = 15.0734
- Upper bound = 15.14 + 0.0666 = 15.2066
Thus, the 99% confidence interval for the mean length cut by the machine is approximately:
\[ 15.0734 \text{ to } 15.2066 \]
Rounding to four decimal places, it becomes:
15.0735 to 15.2065
So the correct choice from the options provided is:
15.0735 to 15.2065.