These data give the diameter (in thousandths of an inch) of motor shafts that will be used in automobile engines. Each day for 80 weekdays, five shafts were sampled from the production line and carefully measured. For these shafts to work properly, the diameter must be about 815 thousandths of an inch. Engineers designed a process that they claim will produce shafts of this diameter, with s = 1 thousandth of an inch.12
(a) If the diameters of shafts produced by this process are normally distributed, then what is the probability of finding a shaft whose diameter is more than 2 thousandths of an inch above the target?
(b) If we measure 80 shafts independently, what is the probability that the diameter of every shaft is between 813 and 817 thousandths of an inch? Between 812 and 818 thousandths?
(c) Group the data by days and generate X-bar and S-charts, putting the limits at ±3 SE. Is the process under control?
(d) Group the data by days and generate X-bar and S-charts, putting the control limits at ±2 SE. Is the process under control?
(e) Explain the results of parts (c) and (d). Do these results lead to contradictory conclusions, or can you explain what has happened?
(f) Only five shafts are measured each day. Is it okay to use a normal distribution to set the control limits in the X-bar chart? Do you recommend changes in future testing?