A mechanical assembly (Figure 7.12) consists of a shaft with a bearing at each end. The total length of the assembly is the sum X + Y + Z of the shaft length X and the lengths Y and Z of the bearings. These lengths may vary from part to part in production, independently of each other and with normal distributions. The shaft length X has mean 11.8 inches and standard deviation of 0.002 inch, while each bearing length Y and Z has a mean 0.4 inch and standard deviation 0.001 inch.
(a) According to the 68-95-99.7 rule about 95% of all shafts have lengths in the range 11.8 d1 inches. What is the value of d1?
Similarly, about 95% of the bearing lengths fall in the range of 0.4 d2. What is the value of d2?
(b) It is a common practice in the industry to state the "natural tolerance" of parts in the form used in part (a). An engineer who know no statistics thinks that tolerances add, so that the natural tolerance for the total length of the assembly (shaft and two bearings) is 12 d inches, where d = d1 + 2 d2. Find the standard deviation of the total length X + Y + Z.
Then find the value of d such that about 95% of all assemblies have lengths in the range of 12 d. Was the engineer correct?