Handbook of Civil Engineering Calculations

Calculation Procedure:

1. Compute the true properties of the sampling distribution
   of the proportion
   Apply Eqs. a and c of the previous calculation procedure, giving fip9 = 0.05 and CT^ =
   (0.05)(0.95)/250 = 0.0001900. Then orp = 0.0138. The sampling distribution diagram ap-
   pears in Fig. 256.


2. Compute the probability that the shipment will be accepted
   The shipment will be accepted if/? < 0.044, and it is necessary to calculate the probability
   of this event. In Fig. 256, at B', z = (0.044 - 0.05)70.0138 = -0.435. From the table of ar-
   eas under the normal curve, A(Z) = 0.168. Then the area to the left of B' = 0.5 - 0.168 =
   0.332. Thus, there is a probability of 33.2 percent that the shipment will be accepted.

Reliability

Consider that a device operates continuously and fails abruptly. The life span of the de-
vice is a continuous random variable. The reliability of the device corresponding to a giv-
en length of time t, denoted by R(t), is the probability that its life span will exceed t. Let
T= life span. Then R(t) = P(T > t).
Refer to Fig. 26, which is the assumed life-span curve of a device. The diagram is con-
structed so that P(J 1 < T ^ t 2 ) = area under curve from tl to t 2. Thus, a life-span curve is
the probability curve of the continuous variable T. Left/(f) = ordinate of life-span curve =
probability-density function. From the definition of reliability, it follows that R(t) = area
under curve to right of t, or

(o=r/(oJt (i6«)

and

dR(f)

f(t)=

dT

(16Z

°

A mechanism formed by the assemblage of devices is called a system, and the indi-

vidual device is called a component of the system. Assume that a system consists of two

FIGURE 26. Life-span curve.

Time.t