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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 F7-MS-Ch9 - 1 F7 Mathematics and Statistics Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution F7-MS-Ch17-2 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. F7-MS-Ch9 - 2 F7 Mathematics and Statistics Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution F7-MS-Ch17-3 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 F7-MS-Ch9 - 3 2 3 4 F7 Mathematics and Statistics Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution F7-MS-Ch9 - 4 F7-MS-Ch17-4 F7 Mathematics and Statistics Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution F7-MS-Ch17-5 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. F7-MS-Ch9 - 5 F7 Mathematics and Statistics Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution F7-MS-Ch17-6 (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 F7-MS-Ch9 - 6 F7 Mathematics and Statistics Chapter 17 Comparison of observed frequency distribution with fitted frequency distribution F7-MS-Ch17-7 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. F7-MS-Ch9 - 7