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 48% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 22strikes. 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

here p=0.48 and n=22

therefore np and n(1-p) both are greater then 10

hence we can use normal approximation of binomial distirbution

here mean number of fish caught =np =22*0.48 =10.56

and std deviaiton =(np(1-p))1/2 =2.3433

a) probability of  12 or fewer fish were caught =P(X<=12) =P(Z<(12.5-10.56)/2.3433)=P(Z<0.8279)

=0.7961

b)P(5 or more fish were caught) =P(X>=5)=1-P(X<=4) =1-P(Z<(4.5-10.56)/2.34)=1-P(Z<-2.5861)

=1-0.0049 =0.9951

c)

between 5 and 12 fish were caught =P(5<=X<=12) =P((4.5-10.56)/2.34<Z<(12.5-10.56)/2.34)

=P(-2.5861<Z<0.8279)=0.7961-0.0049 =0.7913

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