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F7 Mathematics and Statistics
Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution
F7-MS-Ch17-1
SECTION17.1 GENERAL CONSIDERATION
Aim:
1.
2.
3.
4.
5.
to link the two types of distributions together by fitting theoretical distribution to
empirical ones, and then compare the distributions to see how good the fit is.
General Procedure
Identify an appropriate theoretical distribution for the data
Estimate the unknown parameter(s) in the model
Calculate the class probabilities
Calculate the expected class frequencies
Compare the class frequencies
SECTION 17.2
FITTING A DISCRETE UNIFORM DISTRIBUTION
SECTION 17.3
FITTING A POISSON DISTRIBUTION
[see p.386, example9.3]
SECTION 17.4
FITTING A BINOMIAL DISTRIBUTION
[see p.9.4, example9.4]
SECTION 17.5
FITTING A NORMAL DISTRIBUTION
[see p.400, example9.5]
Example
1. Groups of six people are chosen at random and the number, x of people in each group
who wear glasses is record. The results obtained from 200 groups of six are recorded.
Assuming that the situation can be modelled by a binomial distribution having the same
mean as the one calculated from the record.
Calculated the theoretical frequencies and complete the table.
No. in group
wearing glasses
0
1
2
3
4
5
6
No. of groups
observed
frequency
17
53
65
45
18
2
0
Theoretical
frequency
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Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution
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Example
2.
The number of accidents per day was recorded in a district for a period of 1500 days and
the following results were obtained. By fitting a suitable distribution and complete the
table with the theoretical frequency.
No. of accidents
per day
0
1
2
3
4
5
Observed
frequency
342
483
388
176
111
0
Theoretical
frequency
Example
3. The table below gives the number of thunderstorms reported in a particular summer
Number of
Number of
month by 100 meteorological station.
thunderstorms
stations
0
22
(a) Test whether these data may be reasonably regarded as
1
37
conforming a Poisson. Would you expect that these
2
20
3
13
data fit a binomial distribution?
4
6
5
2
(b) Use the mean of the observed frequency to establish the
parameter of a Poisson
distribution and compare with a Poisson distribution with average number of
thunderstorms per month is 1. Compare these two distributions and find out which fit these
data better.
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Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution
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Example
4.
In a large batch of items from a production line the probability that an item is faulty is
p . 400 samples, each of size 5, are taken and the number of faulty items in each batch is
noted. From the frequency distribution below estimate p and work out the expected
frequencies of faulty items per batch for a theoretical binomial distribution having the
same mean.
No. of faulty
items
0
1
2
3
4
5
Observed
frequency
297
90
10
2
1
0
Theoretical
frequency
Example
5. A gardener sows 4 seeds in each of100 plant pots. The number of pots in which r of
the 4 seeds germinate is given in the table below.
No. of seeds germinating
No. of pots
0
13
1
35
2
34
3
15
4
3
Estimate the probability p of an individual seed germinating. Fit a binomial distribution
and find the theoretical frequency.
No. of seeds germinating
Theoretical frequency
0
1
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Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution
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Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution
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C.W.
1)
Eggs are packed in boxes. Each box contains 10 eggs. Tables 1 shows the frequency
distribution of rotten eggs in each of the 120 randomly selected boxes.
(i) It is suggested that the number of the rotten eggs could be modelled by a binomial
distribution with the probability that an eggs is rotten being 0.15. Fill in the missing
expected frequencies, in the third column of Table 1.
(ii) A buyer claims that the number of rotten eggs could be approximated by a Poisson
distribution with mean  . He has calculate some expected frequencies as shown in
Table 1.
(a) Find  , correct to 1 decimal place.
(b) Fill in the missing expected frequencies, in the fourth column of Table 1.
Table 1 Observed and expected frequencies of rotten eggs in each of 120 randomly selected
boxes.
Number of Rotten Observed Frequency Expected Frequency*
Expected Frequency*
Eggs
Binomial
Poisson
0
35
23.6
32.7
1
42
41.7
2
28
27.6
3
11
15.6
4
3
4.8
5
1
1.0
6
0
7 or more
0
0
*correct to 1 decimal place
(iii) The buyer compares the two distributions in (i) and (ii) and adopts the one which fits the
observed data better . He classifies a box of eggs as good if there are less than 2 rotten
eggs in the box. He buys 5 boxes of eggs.
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Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution
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(a) Find the probability that at least 4 of the these boxes are good.
(b) Suppose that at least 4 of these 5 boxes are good and the buyer buys 5 more
boxes. Find the probability that exactly 9 of these 10 boxes are good.
2)
Two machines X and Y fill boxes with detergent powder. Machine X is set to fill each
box with  o kg of the powder. The net weight varies according to a normal distribution ,
with a standard deviation of 0.15kg. Machine Y is set to fill each box with  1kg of the
powder. The net weight also varies according to a normal distribution, but with a
standard deviation of 0.1kg.
(i)
It is desired that no more than 2.5% of the boxes filled by each machine should weigh
less than 2.854kg net. Determine the minimum values of  o and  1.
(ii) The minimum values of  o and  1 determined in (i) are used for production. A
shipment of the product was filled by one and the same machine, but it is not known
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Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution
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whether the machine was X or Y. In order to determine the source of the shipment, a
random sample of 50 boxes of detergent powder was taken and their weights in kg were
recorded as follows:
Table1
Weight in kg
2.7-2.8
2.8-2.9
2.9-3.0
3.0-3.1
3.1-3.2
3.2-3.3
3.3-3.4
Observed Frequency Expected Frequency Expected Frequency
1
4
10
17
13
4
1
Machine X
0.4
2.0
Machine Y
19.2
12.1
3.0
0.3
(a) Fill in the expected frequencies in Table 1.
(b) Compare the observed and the two expected frequency distributions by drawing
histograms. Hence determine the source of the shipment.
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