Sums of Normal random variables. Continue your work in the previous exercise. Dimensions of mechanical parts are often roughly Normal. According to the 68-95-99.7 rule, 95% of rods have lengths within ±d1 of 12 cm and 95% of bearings have lengths within ±d2 of 2 cm.
(a) What are the values of d1 and d2? These are often called the "natural tolerances" of the parts.
(b) Statistical theory says that any sum of independent Normal random variables has a Normal distribution. So the total length of the assembly is roughly Normal. What is the natural tolerance for the total length? It is not d1 + 2d2, because standard deviations don't add.