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:

Question

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

Answer 1

mean=54

variance=.2(1-.2)270=.16*270=43.2
standard deviation:sqrt(43.2)=6.8

does that help?

Answer 2

Thanks.

My solution is:
n=270
p=0.2
np=54
np(1-p)=43.2
sqrt(np(1-p))=sgrt43.2=6.57
z=90-54/6.57=5.479
I don't know how to find the probability that at least 90 people are colour blind, and probability that exactly 50 people are colour blind.
P[x>90]
P[x=50]
Thanks.

Answer 3

To find the probability using the normal approximation in this scenario, we will need to assume that the distribution of color blindness follows a normal distribution. This assumption is valid because the sample size is sufficiently large.

To solve this problem, we need to calculate the mean (μ) and the standard deviation (σ).

Let's start by finding the mean μ:
μ = Sample size (n) × Proportion of color blindness
= 270 × 0.20
= 54

Next, let's calculate the standard deviation σ:
σ = √(Sample size (n) × Proportion of color blindness × (1 - Proportion of color blindness))
= √(270 × 0.20 × 0.80)
≈ 6.36

Now, we can solve the two parts of the question:

a) To find the probability that at least 90 people are color blind, we need to find the cumulative probability from 90 to the upper limit (270):

P(X ≥ 90) = 1 - P(X < 90)

To calculate this probability using the normal approximation, we need to standardize the value using the z-score formula:

z = (X - μ) / σ

For X = 90:
z = (90 - 54) / 6.36
z ≈ 5.66

Using a standard normal distribution table or a calculator, we can find the cumulative probability corresponding to z ≈ 5.66. Subtracting this value from 1, we get the probability that at least 90 people are color blind.

b) To find the probability that exactly 50 people are color blind, we need to calculate the probability at that specific value:

P(X = 50)

Again, we can standardize the value using the z-score formula:

z = (X - μ) / σ

For X = 50:
z = (50 - 54) / 6.36
z ≈ -0.63

Using a standard normal distribution table or a calculator, we can find the probability corresponding to z ≈ -0.63.

Note: The calculations for both scenarios involve finding the area under the normal curve using z-scores. You can use statistical software or online calculators to obtain precise values.