To calculate the 99% confidence interval for the mean length of sheet insulation cut by the machine, we can use the formula for the confidence interval of the mean:
\[ \text{CI} = \bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \(\bar{x}\) is the sample mean
- \(z\) is the z-value for the desired confidence level
- \(\sigma\) is the population standard deviation
- \(n\) is the sample size
Given:
- \(\bar{x} = 15.14\) feet
- \(\sigma = 0.2\) feet
- \(n = 60\)
- For a 99% confidence interval, the z-value is approximately 2.576 (you can find this value in z-tables or using normal distribution calculators).
Now, let's calculate the margin of error (ME):
\[ \text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}} = \frac{0.2}{\sqrt{60}} \approx \frac{0.2}{7.746} \approx 0.0258 \]
Now we calculate the margin of error:
\[ \text{Margin of Error (ME)} = z \times SE = 2.576 \times 0.0258 \approx 0.0665 \]
Now we can calculate the confidence interval:
\[ \text{Lower Limit} = \bar{x} - ME \approx 15.14 - 0.0665 \approx 15.0735 \] \[ \text{Upper Limit} = \bar{x} + ME \approx 15.14 + 0.0665 \approx 15.2065 \]
Thus, the 99% confidence interval for the mean length cut by the machine goes from approximately 15.0735 feet to 15.2065 feet.