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 44% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 26 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

Checking the condition for Normal approximation

np and nq > 5

np = 26 * 0.44 = 11.44 > 5

nq = 26 * (1 - 0.44 ) = 14.56 > 5

Hence we can use Normal approximation to Binomial

Mean = n * P = ( 26 * 0.44 ) = 11.44
Variance = n * P * Q = ( 26 * 0.44 * 0.56 ) = 6.4064
Standard deviation = = 2.5311

Part a)

P ( X >= 12 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 12 - 0.5 ) =P ( X > 11.5 )

P ( X > 11.5 ) = 1 - P ( X < 11.5 )
Standardizing the value

Z = ( 11.5 - 11.44 ) / 2.5311
Z = 0.02

P ( Z > 0.02 )
P ( X > 11.5 ) = 1 - P ( Z < 0.02 )
P ( X > 11.5 ) = 1 - 0.508
P ( X > 11.5 ) = 0.492

Part b)

P ( X >= 5 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 5 - 0.5 ) =P ( X > 4.5 )

P ( X > 4.5 ) = 1 - P ( X < 4.5 )
Standardizing the value

Z = ( 4.5 - 11.44 ) / 2.5311
Z = -2.74

P ( Z > -2.74 )
P ( X > 4.5 ) = 1 - P ( Z < -2.74 )
P ( X > 4.5 ) = 1 - 0.0031
P ( X > 4.5 ) = 0.9969

Part c)

P ( 5 < X < 12 )
Using continuity correction
P ( n + 0.5 < X < n - 0.5 ) = P ( 5 + 0.5 < X < 12 - 0.5 ) = P ( 5.5 < X < 11.5 )

P ( 5.5 < X < 11.5 )
Standardizing the value

Z = ( 5.5 - 11.44 ) / 2.5311
Z = -2.35
Z = ( 11.5 - 11.44 ) / 2.5311
Z = 0.02
P ( -2.35 < Z < 0.02 )
P ( 5.5 < X < 11.5 ) = P ( Z < 0.02 ) - P ( Z < -2.35 )
P ( 5.5 < X < 11.5 ) = 0.5095 - 0.0095
P ( 5.5 < X < 11.5 ) = 0.5