A major automobile company is interested in reducing the time that customers have to wait while having their car serviced with one of the dealers. They select four customers randomly each day and find the total time that each of those customers has to wait (in minutes) while having his or her car serviced. Next, from these four observations, the sample average and range are found. This process is repeated for 25 days. The summary data for these observations are

Σ²⁵_i=1 X^‾_i ^{= 1000,}

Σ²⁵_i=1 R_i = 250

(a) Find the X^‾- and R-chart control limits.

(b) Assuming that the process is in control and that a desirable value on the upper bound of the waiting time is 50 minutes, calculate a process capability index and comment on its value.

(c) Assuming a normal distribution of waiting times, find the proportion of customers that will have to wait more than 50 minutes.

(d) To reduce the waiting time of customers, the service manager hires some additional mechanics, which reduces the average waiting time to 35 minutes. What proportion of the customers will still have to wait more than 50 minutes if the variability in service times is the same as before?