Related Calculations: If the null hypothesis is accepted when, in fact, it is false,
a Type II error has been committed, and the probability of committing this error is denot-
ed by /3. Thus, this calculation procedure involves the determination of'/3. It follows that 1
- /3 is the probability that a false null hypothesis will be rejected. The process of drawing
and analyzing a sample represents a test of the null hypothesis, and the quantities ft and 1
- ft are called the operating characteristic and power, respectively, of the test. Thus, the
power of a test is its ability to detect that the null hypothesis is false if such is truly the
case.
Consider that a diagram is constructed in which assumed values of the parameter are
plotted on the horizontal axis and the corresponding values of ft resulting from the null
hypothesis are plotted on the vertical axis. The curve thus obtained is called an operating-
characteristic curve. Similarly, the curve obtained by plotting values of 1 - ft against as-
sumed values of the parameter is called a power curve.
DECISION BASED ON PROPORTION
OFSAMPLE
A firm receives a large shipment of small machine parts, and it must determine whether
the number of defectives in a shipment is tolerable. Its policy is as follows: A shipment is
accepted only if the estimated incidence of defectives is 3 percent or less, the decision is
based on an inspection of 250 parts selected at random, and a shipment is rejected only if
there is a probability of 90 percent or more that the incidence of defectives exceeds 3 per-
cent. What is the highest incidence of defectives in the sample if the shipment is to be
considered acceptable?
Calculation Procedure:
- Formulate the null hypothesis
Consider that a set of objects consists of type A and type B objects. The ratio of the num-
ber of type A objects to the total number of objects is called the proportion of type A ob-
jects. Let P and p = proportion of type A objects in the population and sample, respective-
ly.
In this case, the population consists of all machine parts in the shipment, the sample
consists of the 250 parts that are inspected, and interest centers on the proportion of de-
fective parts. To provide a basis for investigation, assume that the proportion of defec-
tives in the shipment is precisely 3 percent. Thus, the null hypothesis is H 0 : P = 0.03.
- Compute the properties of the sampling distribution of the
proportion as based on the null hypothesis
Consider that all possible samples of a given size are drawn and their respective values of
p determined, thus obtaining the sampling distribution of p. As before, let N and n = num-
ber of objects in the population and sample, respectively. The sampling distribution of p
has these values: the mean fip = P, Eq. a; the variance (Tp=P(I- P)(N- n)/[n(N- I)], Eq.
b. Where the population is infinite, (Tp=P(I -P)/«, Eq. c. In the present case, P = 0.03, N
may be considered infinite, and n = 250. By Eq. a, /JLP = 0.03; by Eq. c, a* =
(0.03)(0.97)/250 - 0.0001164. Then standard deviation o-p = 0.0108.
- Compute the critical value of p
For simplicity, treat the number of defective parts as a continuous rather than a discrete
