Bert and Ernie noticed that the following are satisfied when Cookie Monster eats

Question

Bert and Ernie noticed that the following are satisfied when Cookie Monster eats cookies: MONSTER (a) the number of cookies eaten during non-overlapping time intervals are independent (b) the probability of exactly one cookie eaten in a sufficiently short interval of length h is approximately Mh; (c) the probability of two or more cookies eaten in a sufficiently short interval is essentially zero Therefore, X1, the number of cookies eaten by Cookie Monster by time t, is a Poisson process, and for any t 0, the distribution ofX is Poisson (Mt) However, Bert and Ernie could not agree on the value of M, the average number of cookies that Cookie Monster eats per minute. Bert claimed that it equals 3, but Ernie insisted that it has been less than 3 ever since Cookie Monster was forced to eat broccoli and carrots. Thus, the two friends decided to test Ho: 3 vs. H 1 3 Bert decided to count the number of cookies Cookie Monster would eat in 4 minutes, X4, and then Reject Ho if X 4 is too small. Ernie, who was the less patient of the two, decided to note how much time Cookie Monster would need to eat the first 6 cookies, T6, and then Reject Ho if T is too large.

Explanation / Answer

Solution

Back-up Theory

Property of Poisson Distribution

If X = number of times an event occurs during period t,

Y = number of times the same event occurs during period kt, and X ~ Poisson(), then

Y ~ Poisson (k) …………………………………………………………………………….. (1)

Now, to work out solution,

Given X = Number of cookies eaten by the Cookie Monster in a minute, then X ~ Poisson ().

If Y = Number of cookies eaten by the Cookie Monster in 4 minutes, then Y ~ Poisson (4)

Given H0 : = 3    Vs HA : < 3

Rejection criterion: Y is very small.

Part (a)

The task is to find the value of ‘very small’, say t, so that level of significance is closest to 0.05. i.e.,

P(Y< t) = 0.05. Using Excel Function for Poisson (12), P(Y < 7) = 0.046, P(Y < 8) = 0.090 and P(Y < 6) = 0.0203.

P(Y < 7) = 0.046 being closest to 0.05, the appropriate test would be:

Reject H0, if the number of cookies eaten by the Cookie Monster in 4 minutes is less than 7. ANSWER

Part (b)

Power of test, at = 2, = 1 – P(Type II error) = 1 – P(accepting H0 when HA is in fact true.)

= 1 – P(Y 7/ = 2 or 4 = 8) = P(Y<7/4 = 8) = 0.313 ANSWER 1

Power of test, at = 1 = 1 – P(Y 7/ = 1 or 4 = 4) = P(Y< 7/4 = 4) = 0.889 ANSWER 2

Part (c)

Level of significance = P(Type I error) = P(rejecting H0 when H0 is in fact true.)

= P(Y 7/ = 3 or 4 = 12) = 0.090 ANSWER

Part (d)

p-value = P(Y 9/ = 3 or 4 = 12) = 0.242 ANSWER

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