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CHAPTER 18
COMPARISON OF EMPIRICAL FREQUENCY
DISTRIBUTIONS WITH FITTED DISTRIBUTIONS
EXERCISE 18.2
Section 18.2
Fitting a discrete uniform distribution
(page 504)
1.
If the 4 models are equally popular, the expected frequency of each cell
=
280
×
1
4
=
70
Model
f
o
f
e
f
o

f
e
2door Coupe
65
70

5
3door Liftback
60
70

10
4door Sedan
80
70
10
5door Liftback
75
70
5
The observed frequencies do not deviate very much from the expected frequencies. Therefore,
there is no evidence to believe that the 4 models are not equally popular.
2.
If the absences occur equally likely on the five weekdays, the probability of an absence should be
5
1
for each weekday. Hence, the expected number of absences should be 100
×
5
1
= 20 for each
day. We tabulate the results as follows:
Day of the
week
Observed
number of
absences,
f
o
Expected
number of
absences,
f
e
Absolute
discrepancy,

f
o
–
f
e

Mon
29
20
9
Tue
12
20
8
Wed
9
20
11
Thu
23
20
3
Fri
27
20
7
We see that the absolute values of the discrepancies are quite large (ranging from
20
3
to
20
11
, i.e.
15% to 55% of the expected frequencies). Hence, we conclude that the absences do not occur
equally likely on the five weekdays.
3.
The total number of customers
= 378 + 331 + 446 + 439 + 417
= 2 011
If the 5 movies attract the same proportion of audience, the number of customers in each studio
should be 2 011
×
5
1
= 402.2. We calculate the discrepancies of the observed and expected data
and tabulate them as follows:
66
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HAPTER
18
C
OMPARISON
OF
E
MPIRICAL
F
REQUENCY
D
ISTRIBUTIONS
WITH
F
ITTED
D
ISTRIBUTIONS
Studio
Observed number
of customers,
f
o
Expected number
of customers,
f
e
Discrepancy,
f
o
–
f
e
A
378
402.2
–24.2
B
331
402.2
–71.2
C
446
402.2
43.8
D
439
402.2
36.8
E
417
402.2
14.8
Total
2 011
2 011.0
0
As the discrepancies are numerically fairly large compared with the expected number of
customers, the manager would not consider each movie as equally popular.
4.
If the die is fair, the probability that each point will occur
=
6
1
.
Hence, the expected frequency of each cell
=
300
×
6
1
=
50
We compare the observed frequencies and the expected frequencies in the table below.
Score
Observed
frequency,
f
o
Expected
frequency,
f
e
Discrepancy,
f
o

f
e
1
47
50

3
2
52
50
2
3
48
50

2
4
57
50
7
5
56
50
6
6
40
50

10
Since the discrepancies are not unreasonably large, the die cannot be considered as biased.
5.
The total number of students,
N
=
164 + 159 + 140 + 150 + 87
=
700
If the data follow the stated uniform distribution,
the expected frequency of each cell
= 700
×
10.5
5
.
20
2

=
140
Age (Years)
f
o
f
e
f
o

f
e
11

12
164
140
24
13

14
159
140
19
15

16
140
140
0
17

18
150
140
10
19

20
87
140

53
Since the discrepancies of the first 4 cells are nonnegative while the discrepancy of the last cell
is a negative and numerically large, the data do not indicate that they follow the stated uniform
distribution.
67
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This note was uploaded on 10/17/2011 for the course IELM 3010 taught by Professor Fugee during the Winter '11 term at HKUST.
 Winter '11
 fugee

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