Fundamentals of Probability and Statistics for Engineers

With intervals Ai defined as shown in the first column of Table 10.6, theoretical

probabilities P(Ai) now can be calculated with the aid of Table A.3. For

example, the first two of these probabilities are

The information given above allows us to construct Table 10.6. Hence,

we have

The number of degrees of freedom in this case is k r 1 6 2 1 3.

Table A.5 thus gives

Since d^2 normal distribution N(83 6, 777 4) is acceptable at the 5%

significance level.

Before leaving this section, let us remark again that statistic D in the^2 test is

(^2) -d istributed only when n. It is thus a large sample test. As a rule, n 50
is considered satisfactory for fulfilling the large-sample requirement.
Table 10.6 Table for^2 test for Example 10.4
x 56 13 0.161 11.27 15.00
56 < x 72 10 0.178 12.46 8.03
72 < x 88 20 0.224 15.68 25.51
88 < x 104 13 0.205 14.35 11.78
104 < x 120 8 0.136 9.52 6.72
120 < x 6 0.096 6.72 5.36
70 1.0 70 72.40
326 Fundamentals of Probability and Statistics for Engineers

P…A 1 †ˆP…X 56 †ˆPU

56  83 : 6



777 : 4

p



ˆFU… 0 : 990 †

ˆ 1 FU… 0 : 990 †ˆ 1  0 : 8389 ˆ 0 : 161 ;

P…A 2 †ˆP… 56 <X 72 †ˆP… 0 : 990 <U 0 : 416 †

ˆ‰ 1 FU… 0 : 416 †Š‰ 1 FU… 0 : 990 †Š

ˆ 0 : 339  0 : 161 ˆ 0 : 178 :

dˆ

Xk

iˆ 1

n^2 i

npi

nˆ 72 : 40  70 ˆ 2 : 40 :

ˆˆ

^23 ; 0 : 05 ˆ 7 : 815 :

<

!1 >

Interval,Ai ni pi npi n^2 i/npi











3, 0 05

..