1. Chapter 9 Comparison of Empirical Frequency Distributions with Fitted Distributions

2. No. in group wearing glasses
No. of groups observed
frequency
Theoretical 17 1 53 2 65 3 45 4 18 5 6
Example
Groups of six people are chosen at random and the number, 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.

3. Example
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
Observed frequency
Theoretical
frequency
342 1
483 2
388 3
176 4
111 5

4. Number of thunderstorms
Example
The table below gives the number of thunderstorms reported in a particular summer month by 100 meteorological station.
(a)Test whether these data may be reasonably regarded as conforming a Poisson. Would you expect that these data fit a binomial distribution?
(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.
Number of thunderstorms
Number of stations 22 1 37 2 20 3 13 4 6 5

5. Example
In a large batch of items from a production line the probability that an item is faulty is samples, each of size 5, are taken and the number of faulty items in each batch is noted. From the frequency distribution below estimate and work out the expected frequencies of faulty items per batch for a theoretical binomial distribution having the same mean.
No. of faulty items
Observed frequency
Theoretical
frequency
297 1 90 2 10 3 4 5

6. Example
A gardener sows 4 seeds in each of100 plant pots. The number of pots in which of the 4 seeds germinate is given in the table below.
Estimate the probability of an individual seed germinating. Fit a binomial distribution and find the theoretical frequency.
No. of seeds germinating 1 2 3 4
No. of pots 13 35 34 15
No. of seeds germinating 1 2 3 4
Theoretical frequency