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 39% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 20 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
To determine if it is appropriate to use the normal approximation to the binomial distribution, we need to check if certain conditions are met. These conditions are:
1. The number of trials, n, is large enough compared to the number of successes, np, and the number of failures, n(1-p).
2. Both np and n(1-p) are greater than or equal to 5.
For this problem, n = 20 (number of strikes), p = 0.39 (probability of catching a fish), and q = 1 - p = 0.61 (probability of not catching a fish).
Now let's check if the conditions are met:
1. n * p = 20 * 0.39 = 7.8
n * q = 20 * 0.61 = 12.2
Since both np and nq are greater than 5, the first condition is met.
2. The number of trials, n, is 20.
Since n is reasonably large, the second condition is also met.
Therefore, it is appropriate to use the normal approximation to estimate the probabilities in this problem.
To estimate the probabilities using the normal distribution, we need to calculate the mean (μ) and standard deviation (σ) of the binomial distribution, and then use these values to find the corresponding probabilities using the standard normal distribution.
The mean (μ) of the binomial distribution is given by μ = n * p, and the standard deviation (σ) is given by σ = √(n * p * q).
In our case, μ = 20 * 0.39 = 7.8, and σ = √(20 * 0.39 * 0.61) = 2.138.
(a) To find the probability that 12 or fewer fish were caught, we need to calculate P(X ≤ 12), where X is the random variable representing the number of fish caught.
Using the normal approximation, we can convert this to a z-score by standardizing it as follows:
z = (x - μ) / σ
where x is the number of fish caught.
For P(X ≤ 12), we find the corresponding z-score using the formula above, and then use a standard normal distribution table or calculator to find the probability.
(b) To find the probability that 5 or more fish were caught, we need to calculate P(X ≥ 5), where X is the random variable representing the number of fish caught.
Again, we standardize it using the z-score formula and find the corresponding probability using the standard normal distribution table or calculator.
(c) To find the probability that between 5 and 12 fish were caught, we need to calculate P(5 ≤ X ≤ 12) using the z-score formula and the standard normal distribution table or calculator.
Please note that these calculations are estimates based on the normal approximation and may not be exact.