It is known that approximately 20% of the population is colour blind. In a sample of 270 people, use the normal approximation to find probability that:

a)at least 90 people are colour blind
b)exactly 50 people are colour blind
Thanks

To solve this problem using the normal approximation, we need to first calculate the mean and standard deviation for the binomial distribution.

In this case, we have a binomial distribution with n = 270 (sample size) and p = 0.2 (probability of being color blind). The mean (μ) is calculated as μ = n * p, and the standard deviation (σ) is calculated as σ = sqrt(n * p * (1 - p)).

a) To find the probability that at least 90 people are color blind, we need to find the cumulative probability for 90 or more successes (i.e., P(X >= 90)).

First, let's calculate the mean and standard deviation:
μ = n * p = 270 * 0.2 = 54
σ = sqrt(n * p * (1 - p)) = sqrt(270 * 0.2 * 0.8) ≈ 7.48

Now, we can use the normal distribution to approximate the binomial distribution:
P(X >= 90) ≈ 1 - P(X < 90)

To convert this into a standard normal distribution, we can standardize using z-scores: z = (X - μ) / σ.

Let's calculate the z-score:
z = (90 - 54) / 7.48 ≈ 4.81

Using a standard normal distribution table or a calculator, we can find the cumulative probability to the left of z = 4.81 (P(Z < 4.81) ≈ 1):

P(X >= 90) ≈ 1 - P(X < 90) ≈ 1 - 1 ≈ 0

Therefore, the probability of having at least 90 people color blind in a sample of 270 is approximately 0.

b) To find the probability that exactly 50 people are color blind, we need to find P(X = 50).

Again, let's calculate the mean and standard deviation:
μ = n * p = 270 * 0.2 = 54
σ = sqrt(n * p * (1 - p)) = sqrt(270 * 0.2 * 0.8) ≈ 7.48

To calculate the probability of exactly 50 people being color blind, we need to use the binomial probability formula:

P(X = 50) = (n choose X) * p^X * (1 - p)^(n - X)

P(X = 50) = (270 choose 50) * 0.2^50 * (1 - 0.2)^(270 - 50)

Using a calculator or a statistical software, we can evaluate this expression to find the probability.

Keep in mind that the normal approximation may not always be accurate for small sample sizes, so it's important to consider the conditions for valid approximations when choosing the method to solve this type of problem.