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 calculate the 95% confidence interval for the average torque being applied during the assembly process, we can use the formula:
Confidence Interval = Average Torque ± (Critical Value) * (Standard Deviation / √Sample Size)
Step 1: Calculate the critical value.
The critical value is based on the desired level of confidence and the sample size. For a 95% confidence level and a sample size of 35, we can look up the critical value from the t-distribution table or calculate it using a statistical software. For simplicity, let's assume the critical value is 2.
Step 2: Calculate the standard error.
The standard error is the standard deviation of the sample mean and is calculated by dividing the standard deviation by the square root of the sample size.
Standard Error = Standard deviation / √Sample size
In this case, the standard deviation (σ) is given as 3 lb-ft, and the sample size (n) is 35. Therefore:
Standard Error = 3 lb-ft / √35
Step 3: Calculate the confidence interval.
Confidence Interval = Average Torque ± (Critical Value) * (Standard Error)
Average Torque = 150 lb-ft (given)
Critical Value = 2 (assumed)
Standard Error = 3 lb-ft / √35
Confidence Interval = 150 lb-ft ± (2 * (3 lb-ft / √35))
Now, plug in the values and calculate the confidence interval:
Confidence Interval = 150 lb-ft ± (2 * (3 lb-ft / √35))
Confidence Interval = 150 lb-ft ± (2 * 0.508 lb-ft)
Confidence Interval = 150 lb-ft ± 1.016 lb-ft
Therefore, the 95% confidence interval for the average torque being applied during the assembly process is (148.984 lb-ft, 151.016 lb-ft).