A cosmetics factory owns a machine that automatically fills its best-selling facial cream jars—4 ounces per jar.

The machine is set to have a mean of 4.1 ounces per jar. Given all other factors, the process shifted the mean away from 4.1 by an under fill. The machine has to stop its production. Regardless of the mean amount being dispensed, the standard deviation of the amount dispensed always has a value of 0.22 ounce. A quality control engineer routinely selects 30 jars from the assembly line to check the amount filled.

a. State the appropriate null and alternative hypothesis for the engineer’s checks.

b. Calculate the chances of Type II to occur using alpha = 0.05.