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

Answer 1

Sample size, n = 29

P(strike), p = 0.42

q = 1 - p = 0.58

Normal approximation can be used is np 5 and nq 5

here, np = 29x0.42 = 12.18

nq = 29x0.58 = 16.82

Both np and nq 5

It is appropriate to use the normal approximation to the binomial.

Mean = np = 12.18

Standard deviation =

= 2.6579

a) P(12 or fewer fish were caught) = P(X 12)

= P(Z < (12.5 - 12.18)/2.6579)   (continuity correction of 0.5 is applied)

= P(Z < 0.12)

= 0.5478

b) P(5 or more fish were caught) = P(X 5)

= 1 - P(X < 4.5)         (continuity correction of 0.5 is applied)

= 1 - P(Z < (4.5 - 12.18)/2.6579)

= 1 - P(Z < -2.89)

= 1 - 0.0019

= 0.9981

c) P(between 5 and 12 fish were caught) = P(5 X 12)

= P(X < 12.5) - P(X < 4.5)

= 0.5478 - 0.0019

= 0.5459