Sands, Inc., makes particleboard for the building industry. Particleboard is built by mixing wood chips and resins together and pressing the sheets under extreme heat and pressure to form a 4-feet 8-feet sheet that is used as a substitute for plywood.

The strength of the particleboards is tied to the board’s weight. Boards that are too light are brittle and do not meet the quality standard for strength. Boards that are too heavy are strong but are difficult for customers to handle. The company knows that there will be variation in the boards’ weight. Product specifications call for the weight per sheet to average 10 pounds with a standard deviation of 1.75 pounds. During each shift, Sands employees select and weigh a random sample of 25 boards. The boards are thought to have a normally distributed weight distribution.
If the average of the sample slips below 9.60 pounds, an adjustment is made to the process to add more moisture and resins to increase the weight (and, Sands hopes, the strength).

a. Assuming that the process is operating correctly according to specifications, what is the probability that a sample will indicate that an adjustment is needed?

b. Assume the population mean weight per sheet slips to 9 pounds. Determine the probability that the sample will indicate an adjustment is not needed.

c. Assuming that 10 pounds is the mean weight, what should the cutoff be if the company wants no more than a 5% chance that a sample of 25 boards will have an average weight less than 9.6 lbs?