Calculation Procedure:
- Compute the values of z corresponding to the specified
boundary values of X
If a random variable X is continuous, the probability that X will assume a value between
Xj and Xk is represented graphically by constructing a probability diagram in this manner:
Plot values ofXon the horizontal axis; then construct a curve such that P(Xj <X<Xk) =
area bounded by the curve, the horizontal axis, and vertical lines at Xj and Xk. The ordi-
nate of this curve is denoted by f(X) and is called the probability density function. The to-
tal area under the curve is 1, the probability of certainty.
A continuous random variable has a normal or gaussian probability distribution if the
range of its possible values is infinite and its probability curve has an equation of this
form: f (X) = (\lb^/2^^)Q-<x-^^2 '2b2 where a and b are constants and e = base of natural log-
arithms. Figure 20 is the probability diagram; the curve is bell-shaped and symmetric
about a vertical line through the summit.
Consider that the trial that yields a value of X is repeated indefinitely, generating an in-
finite set of values of X. Let JJL and a = arithmetic mean and standard deviation, respec-
tively, of this set of values. The summit of the probability curve lies at X= p. By symme-
try, the area under the curve to the left and to the right of X =IJL is 0.5.
The deviation of X, from JJL is expressed in standard units in this form: Z 1 = (X 1 - IJL)/cr.
Thus, for JT= 14, z = O; for X= 17, z - (17 - 14)72.5 = 1.20; for ,T= 12, z = (12 - 14)72.5
= -0.80; etc. Record the z values in Table 24.
TABLE 24
X z A(z)
14 O O
17 1.20 0.38493
12 -0.80 0.28814
16.2 0.88 0.31057
10 -1.60 0.44520
FIGURE 20. Curve of normal probability distribution.
