The Tonight Show with Johnny Carson lasted for 30 years and featured movie stars, comedians, animal acts, and Carnac the Magnificent. The first part of the show was reserved for Johnny’s monologue....

The Tonight Show with Johnny Carson lasted for 30 years and featured movie stars, comedians, animal acts, and Carnac the Magnificent. The first part of the show was reserved for Johnny’s monologue. The mean length of a monologue was 12 minutes with standard deviation 45 seconds. Suppose 40 shows are selected at random and the length of each monologue is recorded.

A (destructive) tensile test is standard procedure for testing a cross-wire weld. Suppose a certain weld is designed to withstand a force with mean 0.8 kilonewtons (kN). To check the quality of welds, a manufacturer randomly selects 25 welds (every hour) and performs a tensile test on each weld. If the mean force required (for the 25 welds) is between 0.75 and 0.85 kN, the process is allowed to continue. Otherwise, it is shut down. Suppose the force distribution is normal with standard deviation 0.1 kN.

a. If the true mean is 0.8 kN, what is the probability that the process will be shut down? This probability represents the chance of making one kind of error.

b. If the true mean is 0.82 kN, what is the probability that the process will be allowed to continue? What if m 5 0.84 kN? These probabilities represent the chance of making a different kind of error.

c. Suppose the process is allowed to continue if the mean force required is between 0.76 and 0.84 kN. Answer parts (a) and (b) with this new interval.