Nifty Cup manufactures stainless steel water bottles for joggers and hikers. Recently Nifty Cup has gotten reports of leaks in their Z-700 model water bottle. The difficulty seems to be a defective seam which gives way when the bottle is filled with carbonated beverages. Unfortunately, Nifty Cup's quality control processes were not designed to detect this flaw. Nifty Cup's engineers have developed a test to determine whether or not a Z-700 water bottle has a defective seam. However this technology is not perfect. For a Z-700 water bottle known to have a faulty seam, the test will yield a positive (indicate that the seam is defective) result 90% of the time. On the other hand, for a Z-700 water bottle known to have a good seam the test will yield a positive 20% of the time.

A. A sample of 100 Z-700 bottles were tested and yielded 47 bottles which tested positive for having a defective seam; the remaining 53 tested negative for the defect. Write out a 95% confidence interval for the proportion of all Z-700 bottles that would test positive, were it possible to test all of them.

B. Nifty Cup would like to know what proportion of all Z-700 bottles actually have the defect. Suppose that the sample proportion of positive tests in part a. is equal to the population proportion of Z-700 bottles that would test positive for the defective seam. Under this assumption, what is the proportion of Z-700 bottles that actually have defective seams?