The weight of connecting rods used in an automobile engine is to be closely controlled to minimize vibrations. The specification is that each rod must be 974 ± 1.2 grams. The half-width of the specified interval, namely, 1.2 grams, is known as the tolerance. The manufacturing process at a plant produces rods whose weights are normally distributed with a mean µ of 973.8 grams and a standard deviation σ of 0.32 grams.

a. What proportion of the rods produced by this process will be acceptable according to the specification?

b. The process capability index, denoted by C_p, is given by the formula
C_p= Tolerance/(3*σ)

Calculate C_p for this process.

c. Would you say a larger value or a smaller value of C_p is preferable?

d. The mean of the process is 973.8 grams, which does not coincide with the target value of 974 grams. The difference between the two is the offset, defined as the difference and therefore always positive. Clearly, as the offset increases, the chances of a part going outside the specification limits increase. To take into account the effect of the offset, another index, denoted by C_pk, is defined as
C_pk = C_p -(Offset/(3*σ))

Calculate C_pk for this process.

e. Suppose the process is adjusted so that the offset is zero, and σ remains at 0.32 gram. Now, what proportion of the parts made by the process will fall within specification limits?

f. A process has a C_p of 1.2 and a C_pk of 0.9. What proportion of the parts produced by the process will fall within specification limits?