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. 3 3 4 4 6 6 1 2 2 3 7 7 8 4 6 3 4 4 6 4 6 5 5 5 5 00 00 00 4 4 4 6 7 9 انحنا | ال 4 5 5 7 7 9 9 4 4 3 7 9 Perform x goodness of fit test for Poisson Distribution at a= 0.01 (state the hypothesis first).

Answer 1

Here we have to test the hypothesis,

H_{​​​​​​0} : Given data fits for Poisson distribution.

H_{​​​​​​1} : Given data doesnot fits for Poisson distribution.

Estimate of parameter lambda is mean of the data

Lamda = 5.14

Test statistic value is given as,

Chi square= 3.7020

Here O_{​​​​​i} is the observed frequency from given data

P(x) are Poisson probabilities obtained using Command Poisson.dist

E_{​​​​​​i} are the expected frequencies

(some expected frequencies are less than 5 so they are clubbed with neighbouring expected frequencies.Corronsponding Observed frequencies are also clubbed.)

At 0.01 level of significance critical value is obtained from chi square distribution with n- k-1 degrees of freedom

Here n=7 ( number of cells for (Oi - Ei )^2/Ei is calculated), k= 1 ( number of estimated parameter)

Degrees of freedom= 7-1-1=5

So from chi square table

Critical value= 15.08627 ( CHIQ.INV( 0.99,5) excel commons)

Since chi square value is less than critical value

Hence we accept H_{​​​​​​0} at 0.01 level of significance and conclude that given data fits for Poisson distribution.