In the following problem, check that it is appropriate to use the normal approxi

Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Ocean fishing for billfish is very popular in the Cozumel region of Mexico. In the Cozumel region about 42% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 29 strikes. Find the following probabilities. (Round your answers to four decimal places.)

(a) 12 or fewer fish were caught


(b) 5 or more fish were caught


(c) between 5 and 12 fish were caught

Explanation / Answer

Let X denote the number of fish caught in n strikes, then X follows Binomial with n=29 and p=0.42

Mean number of fish caught =np=29*0.42 =12.18

Standard deviation of number of fish caught =sqrt(np(1-p)) =sqrt(29*0.42*(1-0.42)) =2.66

As np=12.18>5 and n(1-p)=29*(1-0.42)=16.82>5, so normal approximation can be used.

X follows normal with mean=12.18 and standard deviation sigma=2.66

Part-a

Using continuity correction

P(X<=12)=P(X<12.5)

=0.5479, using excel function =NORMDIST(12.5,12.18,2.66,TRUE)

Part-b

P(X>=5)=P(X>4.5)

=1-P(X<=4.5)

=1-0.0019   , using excel function =NORMDIST(4.5,12.18,2.66,TRUE)

=0.9981

Part-c

P(5<X<12)=P(X<12)-P(X<=5)

=P(X<11.5)-P(X<5.5)

=0.3991-0.0060

=0.3931, using excel function =NORMDIST(11.5,12.18,2.66,TRUE) and =NORMDIST(5.5,12.18,2.66,TRUE)

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