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 45% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 27 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

X ~ Bin ( n , p)

Where n = 27 , p = 0.45

Mean = n p = 27 * 0.45 = 12.15

Standard deviation = sqrt [ n p ( 1 - p) ]

= sqrt [ 27 * 0.45 ( 1 - 0.45) ]

= 2.5851

Using normal approximation,

P(X < x) = P(Z < ( x - Mean) / SD)

a)

Using continuity correction,

P(X <= 12) = P(Z < (12.5 - 12.15) / 2.5851)

= P(Z < 0.14)

= 0.5557

b)

P(X >= 5) = P ( Z > ( 5.5 - 12.15) / 2.5851)

= P(Z > -2.57)

= P(Z < 2.57)

= 0.9949

c)

P( 5 <= X <= 12) = P( 4.5 < X < 12.5)

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

= P(Z < ( 12.5 - 12.15) / 2.5851) - P(Z < (4.5 - 12.15) / 2.5851)

= P(Z < 0.14) - P(Z < -2.96)

= 0.5557 - 0.0015

= 0.5542