particular case involved. In practice, common values for are 0.001, 0.01, and
0.05; a value of between 5% and 1% is regarded as almost significant; a value
between 1% and 0.1% as significant; and a value below 0.1% as highly significant.
Let us now give a step-by-step procedure for carrying out the^2 test when
the distribution of a population X is completely specified.
. (^) Step 1: divide range space X into k mutually exclusive and numerically
convenient intervals Ai,i 1, 2,...,k.Letni be the number of sample values
falling into Ai. As a rule, if the number of sample values in any Ai is less than
5, combine interval Ai with either Ai 1 or Ai 1.
. (^) Step 2: compute theoretical probabilities P(Ai)pi,i 1, 2,...,k,bymeans
of the hypothesized distribution.
. (^) Step 3: construct d as given by Equation (10.7).
. (^) Step 4: choose a value of and determine from Table A.5 for the^2
distribution of (k 1) degrees of fr eedom the value of^2 k1,.
. (^) Step 5: reject hypothesis H if d^2 k1,. Otherwise, accept H.
Ex ample 10. 1. Problem: 300 light bulbs are tested for their burning time t (in
hours), and the result is shown in Table 10.1. Suppose that random burning
time T is postulated to be exponentially distributed with mean burning time
1/ 200 hours; that is, 0 005, perhour, and
Test this hypothesis by using the^2 test at the 5% significance level.
Answer: the necessary steps in carrying out the^2 testareindicatedinTable10.2.
The first column gives intervals Ai, which are chosen in this case to be the
intervals of t given in Table 10.1. The theoretical probabilities P(Ai)pi in the
third column are easily calculated by using Equation (10.8). For example,
Table 10.1 Sample values for
Example 10.1
Burning time, t Number
t < 100 121
100 t < 200 78
200 t < 300 43
300 t 58
n 300
320 Fundamentals of Probability and Statistics for Engineers
>
fT
t 0 :005e^0 :^005 t; t 0 :
10 : 8
p 1 P
A 1 Z 100
00 :005e^0 :^005 tdt 1 e^0 :^5 0 : 39 ;p 2 P
A 2 Z 200
1000 :005e^0 :^005 tdt 1 e^1 0 : 39 0 : 24 :