Question

Q5c: The number of cars Avenues has been tabulated by a group of civil engineering students. They have obtained the data in the table below. passing eastbound through the intersection of Mill and University Does the assumption of a binomial distribution seer this process? Use a = appropriate as a probability model for 0.05 Observed Vehicles per Minute Frequency 40 14 41 24 42 57 43 111 44 194 45 256 46 296 47 378 48 250 49 185 50 171 51 150 52 110

Answer 1

ANSWER:

Given that,

To the given data first we have to fit Poisson distribution using the following PMF and expression for expected frequency -.x=0,1,2.... Where 2 = Mean of the distribution P(X=x) = *2* P(X= ) ! N 147830 2976 = 49.674 Expected frequency e = N P(X=i) Where N be the total frequency. The fitted distribution is as follows.

Vehicles per minute(x) 40 41 1 37 Observed Expected frequency frequency (f.) fx Probability (E) 14 560 0.02292826366 24 984 0.027779022 80 2394 0.032854685 95 111 4773 0.037954083 109 194 8536 0.042848485 123 256 11520 0.047299071 136 296 136160.051076888 147 378 | 17766 0.053982901155 250 12000 0.055865622 161 185 9065 0.056634126 163 171 8550 0.056264939 162 150 7650 0.054802116158 110 5720 0.052350837| 151 102 5406 0.049065634 141 96 5184 0.045134985 130 4950 0.040764326 117 81 4536 0.036159456 104 4161 0.031512052 91 3712 0.026988474 78 3599 0.022722493 65 3540 0.018811974 54 3050 | 0.015319133 44 42 2604 0.012273605 35 29 1827 0.009677457 28 18 | 1152 0.007511228 22 15 975 0.005740203 17 2976 147830 59 65

Ho: The assumption of Poisson fit is good. H: The assumption of Poisson fit is not suitable. a: 0.05 Test statistic: Under Ho, Calculated x = $(0;-E,)) - with (n - k-1) DF. * E; * Where k be the number of parameters obtained from the sample. Here k = 1, Since 1 is estimated from the given data.



43 45 136 46 48 50 171 167 51 Vehicles Observed Expected | Chi - Square per frequency frequency contribution minute(x) (0) (E) (Oi - Ei ^2/Ei 40 14 66 40.96969697 24 | 80 39.21 42 57 95 15.2 111 109 0.036697248 44 194 123 40.98373984 256 105.8823529 296 147 151.0272109 47 378 155 3 20.8322581 161 49.19875776 49 185 163 2.969325153 162 0.5 150 158 0.405063291 52 110 151 11.13245033 141 10.78723404 54 96 130 8.892307692 55 90 117 | 6.230769231 104 1 5.086538462 91 3.56043956 58 2.512820513 61 0.246153846 59 54 0.462962963 61 | 50 44 0.818181818 42 | 1.4 63 29 28 0.035714286 18 22 0.727272727 15 17 0.235294118 2976 2632 8 19.3332417 53 102 56 57 64 78 59 60 62 35

Here DF becomes (25-1-1) = 24 DF. X Critical value with 24 DF at 0.05 level of significance = 36.42 Since Calculated x > x* critical value we reject our null hypothesis. ie. The assumption of Poisson fit is not appropriate.