Question

1. A manufacturing process is aiming for a nominal of 78 ounces with the tolerance of 1. Historically, the cost penalty when the process is out of control, was approximately $7.80 each unit. Based on the information and the table below, please calculate the following:

1. the expected value of the quality dimension, that is E(x);
2. the variance of the quality dimension, that is Var(x);
3. the expected deviation of the quality dimension (the difference between the expected value and the nominal), and E(D) = E(x) - T
4. the expected loss of the process per unit, that is EL.

(Note: I could have asked you what the change in the expected loss is when the process is changed to the expected dimension of 78.04, and the variance of 0.45, but I will leave it as your own practice.)

Table 4.1 Quality Dimensions and their Frequency

Quality Dimensions	Frequency
75	30
76	70
77	280
78	500
79	100
80	10
81	10

Answer 1

Let the random variable, x = quality dimension

The expected value and the variance for the random variable x is obtained using the following formula,

From the data values,

x	f	f*x	f*x^2
75	30	2250	168750
76	70	5320	404320
77	280	21560	1660120
78	500	39000	3042000
79	100	7900	624100
80	10	800	64000
81	10	810	65610
Sum	1000	77640	6028900

The expected value is,

The variance is,

The expected deviation is,

The expected loss is,