A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 ft. It is known that the standard deviation in the cutting length is 0.15 ft. A sample of 70 ut sheets yields a mean length of 12.14 ft. Find the 99% confidence interval for the mean length cut by machine

Formula:

CI99 = mean + or - 2.575(sd/√n)
...where + or - 2.575 represents the 99% confidence interval using a z-table, sd = standard deviation, and n = sample size.

I hope this will get you started.

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To find the 99% confidence interval for the mean length cut by the machine, we can use the formula:

Confidence Interval = Sample Mean ± (Z * Standard Error)

Where:
- Sample Mean is the mean length obtained from the sample (12.14 ft in this case).
- Z is the z-score associated with the desired confidence level (99% confidence level corresponds to a z-score of 2.576).
- Standard Error is the standard deviation of the sample mean, which can be calculated as the standard deviation of the population divided by the square root of the sample size.

First, let's calculate the standard error:

Standard Error = Standard Deviation / √(Sample Size)

Standard Deviation = 0.15 ft
Sample Size = 70

Standard Error = 0.15 / √(70)
Standard Error ≈ 0.017901 ft

Now, let's calculate the confidence interval:

Confidence Interval = 12.14 ± (2.576 * 0.017901)

Upper Confidence Limit = 12.14 + (2.576 * 0.017901)
Upper Confidence Limit ≈ 12.14 + 0.046144
Upper Confidence Limit ≈ 12.186144 ft

Lower Confidence Limit = 12.14 - (2.576 * 0.017901)
Lower Confidence Limit ≈ 12.14 - 0.046144
Lower Confidence Limit ≈ 12.093856 ft

The 99% confidence interval for the mean length cut by the machine is approximately 12.093856 ft to 12.186144 ft.