Fundamentals of Probability and Statistics for Engineers

These theoretical probabilities are given in the third column of Table 10.5.
From column 5 of Table 10.5, we obtain

Table A.5 with 0:05 and k r 1 9 degrees of freedom gives

Since d the hypothesized distribution with 9 09 is accepted at
the 5% significance level.

Example 10.4.Problem: based upon the snowfall data given in Problem 8.2(g)
from 1909 to 1979, test the hypothesis that the Buffalo yearly snowfall can be
modeled by a normal distribution at 5% significance level.
Answer: for this problem, the assumed distribution for X, the Buffalo yearly
snowfall, measured in inches, is N(m,^2 ) where m and^2 must be estimated
from the data. Since the maximum likelihood estimator for m and^2 are
respectively, we have

Table 10.5 Table for^2 test for Example 10.3

0 x < 5 9 0.052 5.51 14.70

5 x < 6 7 0.058 6.15 7.97

6 x < 7 13 0.088 9.33 18.11

7 x < 8 12 0.115 12.19 11.81

8 x < 9 8 0.131 13.89 4.61

9 x < 10 9 0.132 13.99 5.79

10 x < 11 13 0.120 12.72 13.29

11 x < 12 10 0.099 10.49 9.53

12 x < 13 5 0.075 7.95 3.14

13 x < 14 6 0.054 5.72 6.29

14 x 14 0.076 8.06 24.32

106 1.0 106 119.56

M odel Verification 325

dˆ

Xk

iˆ 1

n^2 i

npi

nˆ 119 : 56  106 ˆ 13 : 56 :

ˆ ˆ

^29 ; 0 : 05 ˆ 16 : 92 :

<^2 9,0: 05 , ˆ

 



M^ˆX, andc^2 ˆ[-n1)/n]S^2 ,

m^ˆxˆ

1

70

X^70

jˆ 1

xjˆ 83 : 6 ;

b^2 ˆ^69

70

s^2 ˆ

1

70

X^70

jˆ 1

…xj 83 : 6 †^2 ˆ 777 : 4 :

           

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

: