23 It is conjectured that the number of wrong telephone connections is Poisson w
ID: 3244220 • Letter: 2
Question
23 It is conjectured that the number of wrong telephone connections is Poisson with parameter = 0.5. A total of 120 days of observations produced the following results # wrong connections | days observed | true Poisson distr. with 0 71 37 9 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 1.0 3 total 120 Test the null hypothesis Ho that the daily number of wrong telephone connections is indeed Poisson with = 0.5 at significance level = 0.05 against the alternative hypothesis H1 that it is Solution. The formal hypotheses testing is as follows. Ho: the daily number of wrong telephone connections is Poisson with 0.5 H. the daily number of wrong telephone connections is not Poisson with = 0.5 We continue with the above table expanding it and giving it a more formal interpretation: group | # wrong connections | sets of yi's | days 0 no=71 | 0.6065 ni = 37 | 0.3033 n2 = 9 | 0.0758 0 1 121Explanation / Answer
Explanation
As given in the question, q = (1/120)[0,5]{ni2/Pi0} – 120, where
for i = 0, 1, 2, 3, 4, 5; ni = frequency, i.e., days observed for # of wrong connections, i
Pi0 = true Poisson distribution with = 0.5 and 120 is the total frequency.
Step-by-step computations are presented below:
i
ni
ni^2
Pi^0
ni^2/Pi^0
0
71
5041
0.6065
8311.624073
1
37
1369
0.3033
4513.682822
2
9
81
0.0758
1068.601583
3
2
4
0.0126
317.4603175
4
1
1
0.0016
625
5
0
0
0.0002
0
Total
120
1
14836.3688
(1/120) [0,5]{ni2/Pi0} = 14836.3688/120 = 123.6364
q = 123.6364 – 120 = 3.6364.
DONE
i
ni
ni^2
Pi^0
ni^2/Pi^0
0
71
5041
0.6065
8311.624073
1
37
1369
0.3033
4513.682822
2
9
81
0.0758
1068.601583
3
2
4
0.0126
317.4603175
4
1
1
0.0016
625
5
0
0
0.0002
0
Total
120
1
14836.3688
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