Question

2. (25 P) A simulation team collected and sorted the data below for the number of arrivals of customers for a restaurant for 50 days. 1 2 3 1 3 4 6 7 4 2 3 4 6 A 5 2 3 4 6 8 6 3 4 5 6 8 7 4 4 5 6 8 8 4 4 5 6 8 9 2 4 5 7 9 10 3 4 5 7 9 11 3 4 5 7 3 4 5 7 9 12 13 Perform x goodness of fit test for Poisson Distribution at a= 0.01 (state the hypothesis first). 14 15 16

Answer 1

We have to perform chi-square test for goodness of fit to test that poisson distribution is good fit for the data or not at 0.01 level of significance.

Hypothesis -

H₀ - Poisson distribution is good fit for given data.

H₁ - Poisson distribution is not good fit for given data.

Test statistic -

x =Σ (Ο, – Ε)2 Ei

Test criterion -

We have to reject H₀ if $\chi ^{2}$ $\geq$$\chi ^{2}$_{$\alpha$,n-p-k-1}

Where, n - number of classes, p - number of parameters estimated, k - number of pullings

Now, to find the probability for each observation, we have to use PMF of poisson distribution -

e-1 P(X = ) ..!
; x = 0, 1, 2,.............., n ; $\lambda > 0$

= ; otherwise

Now, we first estimate value of $\lambda$ . We know that sample mean is an unbiased estimator of $\lambda$ .

So, $\lambda$ =

T

Observation table -

X	Frequency(Oi)	xf
1	1	1
2	3	6
3	7	21
4	12	48
5	7	35
6	6	36
7	6	42
8	4	32
9	4	36
Total	50	257

Calculations -

257 IS fixi Σfi = 5.14 50

So, $\lambda$ = = 5.14

T

Now, we can calculate probabilities by using below formula -

-5.14 (5.14) I e P(X = 2) C!

Observation table -

X	Frequency(Oi)	probability(Pi)	Ei	Ei	Oi	(Oi-Ei)^2/Ei
1	1	0.0301	1.505	-	-	-
2	3	0.0774	3.87	5.375	4	0.3517
3	7	0.1326	6.63	6.63	7	0.0206
4	12	0.1704	8.52	8.52	12	1.4214
5	7	0.1751	8.755	8.755	7	0.3518
6	6	0.15	7.5	7.5	6	0.3
7	6	0.1102	5.51	5.51	6	0.0436
8	4	0.0708	3.54	5.56	8	1.0708
9	4	0.0404	2.02	-	-	-
Total	50	0.957	47.85	-	-	3.5599

If expected frequency is less than 5, then we have to do pullings by adding above or below expected frequency to that less than 5 expected frequency.

Calculations -

x =Σ (Ο, – Ε)2 = 3.5599 Εί

Critical value -

$\chi ^{2}$_{$\alpha$,n-p-k-1} =  $\chi ^{2}$_{0.01,10-1-2-1} =  $\chi ^{2}$_0.01,6 = 16.812

Where, n - number of classes = 10, p - number of parameters estimated = 1 , k - number of pullings = 2

conclusion -

Since, $\chi ^{2}$ (3.5599) < $\chi ^{2}$ _{$\alpha$,n-p-k-1} (16.812) So, we have to accept H₀ at 0.01 level of significance.

Result -

There is sufficient evidence that poisson distribution is good fit for given data at 0.01 level of significance.

Degrees of Freedom 0.99 0.01 10 11 12 13 14 15 0.975 0.001 0.051 0.216 0484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5812 6408 7.015 7.633 8.260 8.897 9542 10.196 10.856 11 524 12.198 12.879 13.565 14.257 14.954 Chi-Square (1) Distribution Area to the Right of Critical Value 0.95 0.90 0.10 0.05 0.004 0.016 2.706 3.841 0.103 0.211 4.605 5.991 0.352 0.584 6.251 7.815 0.711 1.064 7.779 9.488 1.145 1.610 9.236 11.071 1.635 2.204 10.645 12.592 2.167 2.833 12.017 14.067 2.733 3.490 13.362 15.507 3.325 4.168 14.684 16.919 3.940 4.865 15.987 18.307 4.575 5.578 17.275 19.675 5.226 6.304 18.549 21.026 5.892 7.042 19.812 22.362 6.571 7.790 21.064 23.685 7.261 8.547 22.307 24.996 7.962 9.312 23.542 26.296 8.672 10.085 24.769 27.587 9.390 10.865 25.989 28.869 10.117 11.651 27.204 30.144 10.851 12.443 28.412 31.410 11.591 13.240 29.615 32.671 12.338 14.042 30.813 33.924 13.091 14.848 32.007 35.172 13.848 15.659 33.196 36.415 14.611 16.473 34.382 37.652 15.379 17.292 35.563 38.885 16.151 18.114 36.741 40.113 16.928 18.939 37.916 41.337 17.708 19.768 39.087 42.557 18.493 20.599 40.256 43.773 0.025 5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.194 44.461 45.722 46.979 6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 16 17 18 19 20 21 22 23 24 25 26 29 30