A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired mean length of the insulation is 15 feet. It is known that the standard deviation in the cutting length is 0.2 feet. A sample of 60 cut sheets yields a mean length of 15.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine. True or False: The confidence interval indicates that the machine is not working properly.

1 answer

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.

  1. Identify the sample statistics:

    • Sample mean (\( \bar{x} \)) = 15.14 feet
    • Population standard deviation (\( \sigma \)) = 0.2 feet
    • Sample size (\( n \)) = 60
  2. Calculate the standard error (SE): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.2}{\sqrt{60}} \approx \frac{0.2}{7.746} \approx 0.0258 \]

  3. 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.

  4. Calculate the margin of error (ME): \[ ME = z \times SE = 2.576 \times 0.0258 \approx 0.0664 \]

  5. 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 \( (15.0736, 15.2064) \).

  6. 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.