Page 342 - Fundamentals of Probability and Statistics for Engineers

Model Verification 325

Table 10.5 Table for 2 test for Example 10.3
2
Interval, A i n i p i np i n /np i
i
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

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

k 2
X n i
d ˆ n ˆ 119:56 106 ˆ 13:56:
np i
iˆ1
Table A.5 with ˆ 0:05 and k 1 ˆ 9 degrees of freedom gives
r
2 ˆ 16:92:
9;0:05
Since d < 2 9, 0:05 , the hypothesized distribution with ˆ :
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
^
2
M ˆ X, and ˆ [-n 1)/n]S , respectively, we have
c 2
70
1 X
^ m ˆ x ˆ x j ˆ 83:6;
70
jˆ1
70
69 2 1 X 2
b 2
ˆ s ˆ x j 83:6† ˆ 777:4:
70 70
jˆ1

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