Darlington Industries produces a wide array of home safety and security products, including smoke alarms.

As part of its quality-control program, it constantly tests to assure that the alarms work. A change in the manufacturing process requires the company to determine the proportion of alarms that fail the qualitycontrol tests. Each day, 20 smoke alarms are taken from the production line and tested, and the number of defectives is recorded. A file titled Smokeless contains the results from the last 90 days of testing.

a. Compute the proportion of defective smoke alarms.

b. Calculate the expected number and the standard deviation of defectives for each day’s testing. Assume the proportion of defectives is what was computed in part

a. (Hint: Recall the formulas for the mean and the standard deviation for a binomial distribution.)

c. To make sure that the proportion of defectives does not change, the quality-control manager wants to establish control limits that are 3 standard deviations above the mean and 3 standard deviations below the mean. Calculate these limits.

d. Determine the probability that a randomly chosen set of 20 smoke alarms would have a number of defectives that was beyond the control limits established in part c.