Page 337 - Fundamentals of Probability and Statistics for Engineers

320 Fundamentals of Probability and Statistics for Engineers

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 A i , i ˆ 1, 2, ..., k. Let n i be the number of sample values
falling into A i . As a rule, if the number of sample values in any A i is less than
5, combine interval A i with either A i 1 or A i 1 .
‡

. Step 2: compute theoretical probabilities P(A i ) ˆ p i , i ˆ 1, 2, ..., k, by means
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 freedom the value of 2 .

k 1,
. Step 5: reject hypothesis H if d > 2 . Otherwise, accept H.

k 1,

Example 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/ ˆ 200 hours; that is, ˆ :
0 005, per hour, and
f t†ˆ 0:005 e 0:005t ; t 0: 10:8†
T
Test this hypothesis by using the 2 test at the 5% significance level.
Answer: the necessary steps in carrying out the 2 test areindicatedinTable 10.2.
The first column gives intervals A i , which are chosen in this case to be the
intervals of t given in Table 10.1. The theoretical probabilities P(A i ) ˆ p i in the
third column are easily calculated by using Equation (10.8). For example,
Z 100
p 1 ˆ P A 1 †ˆ 0:005 e 0:005t dt ˆ 1 e 0:5 ˆ 0:39;
0
200
Z
p 2 ˆ P A 2 †ˆ 0:005 e 0:005t dt ˆ 1 e 1 0:39 ˆ 0:24:
100
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 ˆ 300

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