Sorry but I don't know how to do this problem. Can someone please help me?
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 whether it is appropriate to use the normal approximation to the binomial, we can consider the conditions required for the normal approximation. The conditions are:
1. The number of observations, n, is large (typically greater than 30). In this case, the fleet of fishing boats got a total of 20 strikes, which is not a large sample size.
2. The number of successes is a fixed proportion, p, of the total number of observations. In this case, the proportion of strikes resulting in a catch is 39%, which is a fixed proportion.
Since the sample size is not large, it is not appropriate to use the normal approximation to the binomial in this case. Instead, we should use the binomial distribution to find the probabilities.
To find the probabilities, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where n is the number of trials, k is the number of successes, p is the probability of success, and (n choose k) represents the number of combinations of n items taken k at a time.
Now let's proceed to find the requested probabilities:
(a) To find the probability that 12 or fewer fish were caught, we need to calculate the sum of probabilities for k = 0 to 12:
P(X <= 12) = P(X = 0) + P(X = 1) + ... + P(X = 12)
(b) To find the probability that 5 or more fish were caught, we need to calculate the sum of probabilities for k = 5 to 20:
P(X >= 5) = P(X = 5) + P(X = 6) + ... + P(X = 20)
(c) To find the probability that between 5 and 12 fish were caught, we need to calculate the sum of probabilities for k = 5 to 12:
P(5 <= X <= 12) = P(X = 5) + P(X = 6) + ... + P(X = 12)
To calculate these probabilities, we can substitute the appropriate values into the binomial probability formula and calculate each term.
To determine whether it is appropriate to use the normal approximation to the binomial, we need to check if the conditions for using the normal approximation are met. These conditions are:
1. The number of trials, n, is large: n > 10
2. Both np and n(1-p) are greater than 5, where p is the probability of success for each trial.
Let's check these conditions for the given problem:
Given: p = 0.39 (probability of success)
n = 20 (number of trials)
Conditions:
1. n = 20 is greater than 10, so the first condition is satisfied.
2. np = (0.39)(20) = 7.8,
n(1-p) = (20)(1 - 0.39) = 12.2.
Both np and n(1-p) are greater than 5, so the second condition is satisfied.
Since both conditions are met, we can proceed to use the normal distribution to estimate the requested probabilities.
To solve parts (a), (b), and (c), we will use the normal approximation to the binomial distribution. The mean (μ) and standard deviation (σ) of the binomial distribution can be approximated using the formulas:
μ = np
σ = √(np(1-p))
Let's calculate them:
μ = (0.39)(20) = 7.8
σ = √((0.39)(20)(1-0.39)) ≈ 2.159
Now we can use the normal distribution to estimate the probabilities.
(a) To find the probability that 12 or fewer fish were caught, we need to find P(X ≤ 12), where X follows a normal distribution with μ = 7.8 and σ = 2.159.
(b) To find the probability that 5 or more fish were caught, we need to find P(X ≥ 5), where X follows a normal distribution with μ = 7.8 and σ = 2.159.
(c) To find the probability that between 5 and 12 fish were caught, we need to find P(5 ≤ X ≤ 12), where X follows a normal distribution with μ = 7.8 and σ = 2.159.
To calculate these probabilities, we will use a standard normal distribution table or calculator.
Please note that further calculations are needed to find the precise probabilities.