Fundamentals of Probability and Statistics for Engineers

We thus have

These values are indicated in the third column of Table 10.3.
Column 5 of Table 10.3 gives

With k 7, the value of^2 k1,^2 is found from Table A.5 to be

2

Since d^2 6, 01 , the hypothesis is rejected at the 1% significance level.

10.2.2 The Case of Estimated Parameters

Let us now consider a more common situation in which parameters in the
hypothesized distribution also need to be estimated from the data.
A natural procedure for a goodness-of-fit test in this case is first to estimate
the parameters by using one of the methods developed in Chapter 9 and then to
follow the^2 test for known parameters, already discussed in Section 7.2.1. In

Table 10.3 Table for^2 test for Example 10.2

x 0 5147 0.6188 4853 5459

0 < x 1 1859 0.2970 2329 1484

1 < x 2 595 0.0713 559 633

2 < x 3 167 0.0114 89 313

3 < x 4 54 0.0013 10 292

4 < x 5 14 0.0001 1 196

5 < x 6 0.0001 1 36

7842 1.0 7842 8413

Note: ni, observed number of occurrences; pi,

theoretical P(Ai).

322 Fundamentals of Probability and Statistics for Engineers

P…Ai†ˆpiˆ

… 0 : 48 †i^1 e^0 :^48

…i 1 †!

; iˆ 1 ; 2 ;...; 6 ;

P…A 7 †ˆp 7 ˆ 1 

X^6

iˆ 1

pi;

dˆ

Xk

iˆ 1

n^2 i

npi

nˆ 8413  7842 ˆ 571 :

ˆ  ˆ

>

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

     

6, 0 01

6 0 01ˆ16 812

:

;: : :

0 :