To determine whether the confidence interval indicates that the machine is not working properly, we need to calculate the 99% confidence interval for the mean length of the sheet insulation based on the sample data provided.
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Identify the sample statistics:
- Sample mean (\( \bar{x} \)) = 15.14 feet
- Population standard deviation (\( \sigma \)) = 0.2 feet
- Sample size (\( n \)) = 60
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Calculate the standard error (SE): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.2}{\sqrt{60}} \approx \frac{0.2}{7.746} \approx 0.0258 \]
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Determine the critical z-value for a 99% confidence level: The critical z-value for a 99% confidence interval (found from z-tables) is approximately 2.576.
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Calculate the margin of error (ME): \[ ME = z \times SE = 2.576 \times 0.0258 \approx 0.0664 \]
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Construct the 99% confidence interval: \[ CI = \bar{x} \pm ME = 15.14 \pm 0.0664 \]
- Lower limit = 15.14 - 0.0664 = 15.0736
- Upper limit = 15.14 + 0.0664 = 15.2064
Therefore, the 99% confidence interval is approximately \( (15.0736, 15.2064) \).
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Interpret the results: The null hypothesis is that the true mean length of the insulation is 15 feet. The 99% confidence interval from our sample is \( (15.0736, 15.2064) \). Since the entire interval is above 15 feet, we do not have sufficient evidence at the 99% confidence level to conclude that the machine is not working properly.
The correct statement is: False. The confidence interval does not indicate that the machine is not working properly; rather, it suggests that the mean length of the cut sheets is likely above the desired mean of 15 feet, but we cannot definitively conclude a malfunction just from this information alone.
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