Question

1. Fasten-it-All produces wood and metal screws. The screws must be manufactured to within certain tolerances or they are considered defective. Using 99% confidence, a machine that may be producing more than 1.25% defective screws must be shut down. To test Machine 1, a QC inspector randomly samples 1000 screws. The QC inspector’s random sample of 1000 screws contains 7 defective screws.

a. Based on this sample, compute by hand (using the normal approximation to the binomial) and interpret a 99% confidence interval for Machine 1’s defect rate. Confirm your result with Minitab.

b. Based on the 99% confidence interval computed in a, what recommendation would you make regarding whether Machine 1 should be shut down?

c. Suppose the QC inspector had, instead, taken a sample of 2000 screws and found 14 to be defective. Note that this gives the same sample percentage as part a. Based on this sample, compute by hand (using the normal approximation to the binomial) and interpret a 99% confidence interval for Machine 1’s defect rate? Confirm your result with Minitab.

d. Based on the 99% confidence interval computed in c, what recommendation would you make regarding whether Machine 1 should be shut down? Briefly explain (perhaps in 3 words), how the same sample percentage produced different recommendations in parts b and d.  

Answer 1

a. The proportion is

99% confidence interval for Machine 1’s defect rate is

Based on our sample, we are 99% confident that the true percentage of defective screws is between 0.02% and 1.38%

b.

We are not 99% confident that Machine 1 will have a defective rate below 1.25%. Since there is a reasonable chance for the machine to have a defective rate of 1.25%, or more, we should shut down the machine.

c. Suppose the QC inspector had, instead, taken a sample of 2000 screws and found 14 to be defective.

The proportion is

99% confidence interval for Machine 1’s defect rate is

d. Based on the 99% confidence interval computed in c, machine should stay in production since, because the whole interval is 0.21% to 1.18%, we are 99% confident that the defect rate is below 1.25%. Again, it’s a close call, so we will want to regularly test the machine.