In the following problem check that it is appropriate to use

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 46% 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

Solution

Ans: Considering this is a binomial distribution then p = 0.46, q = 0.54, n = 27.
p = probability of success, i.e. a catch.
q = probability of failure
n= number of trials.

Here, the mean is np, the standard deviation is sqrt (npq).
Mean = 27 x 0.46 = 12.42
SD = sqrt(27 x 0.46 x 0.54) = 6.7068

To find the above answers, we need to know 12 and 5 are how distant in terms of units from the mean and standard deviation.

12 is distant 1.23 below the mean, or 0.473 SDs below the mean.
5 is distant 8.23 below the mean, or 3.17 SDs below the mean
So you want the area under the normal curve split into 3 regions:
(a) area under graph that is below 0.473 SD below the mean
(b) area above 3.17 SD below the mean
(c) area between 0.473 and 3.17 SD below the mean

From the tables the values are:
(a) area = 1 - (0.5 + 0.1818) = 0.3182
(b) area = 0.5 + 0.4992 = 0.9992
(c) area = 0.9992 - (0.1818 + 0.5) = 0.3174
The answer is : probability = 0.3174 or about 31.7 per cent.