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9-7 TESTING FOR GOODNESS OF FIT 319

3. H : The form of the distribution is nonnormal.
1
4. 0.05
5. The test statistic is

k 1o
E 2 2
2
0 a i i
i 1 E i
6. Since two parameters in the normal distribution have been estimated, the chi-square
statistic above will have k
p
1 8
2
1 5 degrees of freedom.
2
if 2 11.07.
Therefore, we will reject H 0 0 0.05,5
7. Computations:

8 1o
E 2 2
2
a i i
0 E
i 1 i
2 2 2
112
12.52 114
12.52 114
12.52
p
12.5 12.5 12.5
0.64
2
8. Conclusions: Since 0.64 2 0.05,5 11.07, we are unable to reject H 0 , and there
0
is no strong evidence to indicate that output voltage is not normally distributed. The
2
P-value for the chi-square statistic 0.64 is P 0.9861.
0
EXERCISES FOR SECTION 9-7

9-59. Consider the following frequency table of observa- defined as the number of calls during that one-hour period.
tions on the random variable X. The relative frequency of calls was recorded and reported as

Values 0 1 2 3 4 Value 5 6 8 9 10
Observed Frequency 24 30 31 11 4 Relative
Frequency 0.067 0.067 0.100 0.133 0.200
(a) Based on these 100 observations, is a Poisson distribution
Value 11 12 13 14 15
with a mean of 1.2 an appropriate model? Perform a good-
Relative
ness-of-fit procedure with 0.05.
Frequency 0.133 0.133 0.067 0.033 0.067
(b) Calculate the P-value for this test.
9-60. Let X denote the number of flaws observed on a (a) Does the assumption of a Poisson distribution seem appro-
large coil of galvanized steel. Seventy-five coils are in- priate as a probability model for this data? Use 0.05.
spected and the following data were observed for the values (b) Calculate the P-value for this test.
of X: 9-62. Consider the following frequency table of observa-
tions on the random variable X:
Values 1 2 3 4 5 6 7 8
Observed Values 0 1 2 3 4
Frequency 1 11 8 13 11 12 10 9 Frequency 4 21 10 13 2
(a) Does the assumption of the Poisson distribution seem ap-
(a) Based on these 50 observations, is a binomial distribution
propriate as a probability model for this data? Use 0.01.
with n 6 and p 0.25 an appropriate model? Perform
(b) Calculate the P-value for this test.
a goodness-of-fit procedure with 0.05.
9-61. The number of calls arriving at a switchboard (b) Calculate the P-value for this test.
from noon to 1 PM during the business days Monday through
Friday is monitored for six weeks (i.e., 30 days). Let X be