Fundamentals of Probability and Statistics for Engineers

320 Fundamentals of Probability and Statistics for Engineersparticular case involved. In practice, common values for are 0.001, 0.01, and0.05; a value of between 5% and 1% is regarded as almost significant; a valuebetween 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 whenthe distribution of a population X is completely specified.. Step 1: divide range space X into k mutually exclusive and numericallyconvenient intervals A i ,i ˆ 1,2,...,k. Let n i be the number of sample valuesfalling into A i . As a rule, if the number of sample values in any A i is less than5, 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 meansof the hypothesized distribution.Step 3: construct d as given by Equation (10.7).2Step 4: choose a value of and determine from Table A.5 for the 2distribution of (k 1) degrees of freedom the value of k 1, .. 2Step 5: reject hypothesis H if d > k 1, . Otherwise, accept H.Ex ample 10. 1. Problem: 300 light bulbs are tested for their burning time t (inhours), and the result is shown in Table 10.1. Suppose that random burningtime T is postulated to be exponentially distributed with mean burning time1/ ˆ 200 hours; that is, ˆ 0:005, per hour, andf T …t† ˆ0:005 e 0:005t ; t 0: …10:8†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 A i , which are chosen in this case to be theintervals of t given in Table 10.1. The theoretical probabilities P(A i ) ˆ p i in thethird column are easily calculated by using Equation (10.8). For example,p 1 ˆ P…A 1 †ˆp 2 ˆ P…A 2 †ˆZ 1000Z 2001000:005 e 0:005t dt ˆ 1 e 0:5 ˆ 0:39;0:005 e 0:005t dt ˆ 1 e 1 0:39 ˆ 0:24:Table 10.1 Sample values forExample 10.1Burning time, tNumbert < 100 121100 t < 200 78200 t < 300 43300 t 58n ˆ 300TLFeBOOK