An assembly process includes a torque wrench device that automatically tightens compressor housing bolts; the device has a known process standard deviation of ó = 3 lb-ft in the torque applied. A simple random sample of 35 nuts is selected, and the average torque to which they have been tightened is 150 lb-ft. What is the 95% confidence interval for the average torque being applied during the assembly process?
To determine the 95% confidence interval for the average torque being applied during the assembly process, you can use the formula:
Confidence Interval = X̄ ± Z * (σ/√n)
Where:
X̄ is the sample mean (average torque)
Z is the z-score corresponding to the desired confidence level (in this case, 95% confidence)
σ is the population standard deviation (known process standard deviation)
n is the sample size
In this case, the sample mean (X̄) is given as 150 lb-ft, the known process standard deviation (σ) is 3 lb-ft, and the sample size (n) is 35.
First, find the z-score corresponding to the desired 95% confidence level. The z-score for a 95% confidence level can be obtained from a standard normal distribution table or using statistical software. For a 95% confidence level, the z-score is approximately 1.96.
Next, substitute the given values into the formula:
Confidence Interval = 150 ± 1.96 * (3/√35)
Simplifying the equation:
Confidence Interval = 150 ± 1.96 * (3/√35)
Confidence Interval = 150 ± 1.96 * (0.507)
Calculating the values:
Confidence Interval = 150 ± (0.993)
Therefore, the 95% confidence interval for the average torque being applied during the assembly process is approximately 149.007 lb-ft to 150.993 lb-ft.