Chapter 9 Comparison of Empirical Frequency Distributions with Fitted Distributions

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.

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

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

Example In a large batch of items from a production line the probability that an item is faulty is . 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 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

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