In a book the characters are trying to determine whether the Poisson probabilistic model can be used to describe the locations that rockets landed in a city. They divided the city into

0.25-kmsquared2

regions. They then counted the number of rockets that landed in each region, with the results shown in the table below. Complete parts (a) through (e).

Number of rocket hits	0	1	2	3	4	5	6	7
Observed number of regions	220	209	98	32	8	0	0	1

(a) Estimate the mean number of rocket hits in a region by computing

mu equals Summation from nothing to nothing xP left parenthesis x right parenthesisμ=∑xP(x).

muμequals=0.95070.9507

(Round to four decimal places as needed.)

(b) Explain why the requirements for conducting a goodness-of-fit test are not satisfied.

Select all the reasons why the requirements are not satisfied below.

 

A.

The data are not categorical.

Your answer is not correct.

 

B.

The data are not normally distributed.

Your answer is not correct.

 

C.

Not all expected counts are greater than 1.

This is the correct answer.

 

D.

More than 20% of the expected counts are less than 5.

Your answer is correct.

(c) After consolidating the table, we obtain the following distribution for rocket hits. Using the

Poisson probability model, Upper P left parenthesis x right parenthesis equals StartFraction mu Superscript x Over x exclamation mark EndFraction e Superscript negative muP(x)=μxx!e−μ, where muμ is the mean from part (a), we can obtain the probability distribution for the number of rocket hits. Find the probability of 0 hits in a region. Then find the probability of 1 hit, 2 hits, 3 hits, and 4 or more hits.

Number of rocket hits	0	1	2	3	4 or more
Observed number of regions	220220	209209	9898	3232	99
Probability	0.38650.3865	0.36740.3674	0.17470.1747	0.05530.0553	0.01610.0161

(Round to four decimal places as needed.)

(d)	A total of n equals=568568 rockets was fired. Determine the expected number of regions hit by
	computing "expected number of regions" equals=np, where p is the probability of observing that particular number of hits in the region.

Number of rocket hits	0	1	2	3	4 or more
Observed number of regions	220220	209209	9898	3232	99
Expected number	219.51219.51	208.69208.69	99.2099.20	31.4431.44	9.159.15

(Round to two decimal places as needed.)

(e)	Conduct a goodness-of-fit test for the distribution using the alphaαequals=0.05 level of significance.
	Do the rockets appear to be modeled by a Poisson random variable?

Let

p 0p0,

p 1p1,

p 2p2,

and

p 3p3

be the probability that there are 0, 1, 2, and 3 rocket hits in a given region, and let

p 4p4

be the probability that there are 4 or more. Choose the correct null and alternative hypotheses below.

 

A.

Upper H0:

The rockets strikes can be modeled by a Poisson random variable.

Upper H1:

The rocket strikes do not follow a Poisson distribution.Your answer is correct.

 

B.

Upper H0:

p 1 equals p 2 equals p 3 equals p 4p1=p2=p3=p4

Upper H1:

p 1 not equals p 2 not equals p 3 not equals p 4p1≠p2≠p3≠p4

 

C.

Upper H0:

The rocket strikes do not follow a Poisson distribution.

Upper H1:

The rockets strikes can be modeled by a Poisson random variable.

 

D.

Upper H0:

p 1 not equals p 2 not equals p 3 not equals p 4p1≠p2≠p3≠p4

Upper H1:

p 1 equals p 2 equals p 3 equals p 4p1=p2=p3=p4Compute the test statistic,

chi Subscript 0 Superscript 2χ20.

chi Subscript 0 Superscript 2χ20equals=0.0290.029

(Round to three decimal places as needed.)

Obtain the P-value.

P-value equals=11

(Round to four decimal places as needed.)

State the conclusion.

 

A. Do not reject

Upper H0.

There is sufficient evidence to reject the claim that the rocket strikes can be modeled by a Poisson random variable.

 

B. Reject

Upper H0.

There is sufficient evidence to reject the claim that the rocket strikes can be modeled by a Poisson random variable.

 

C. Reject

Upper H0.

There is insufficient evidence to reject the claim that the rocket strikes can be modeled by a Poisson random variable. Your answer is not correct.

 

D. Do not reject

Upper H0.

There is insufficient evidence to reject the claim that the rocket strikes can be modeled by a Poisson random variable.