Question

Colorblindness is any abnormality of the color vision system that causes a person to see colors differently than most people, or to have difficulty distinguishing among certain colors (www.visionrx.com).

Colorblindness is gender-based, with the majority of sufferers being males.
Roughly 8% of white males have some form of colorblindness, while the incidence among white females is only 1%. A random sample of 20 white males and 40 white females was chosen.
Let X be the number of males (out of the 20) who are colorblind.
Let Y be the number of females (out of the 40) who are colorblind.
Let Z be the total number of colorblind individuals in the sample (males and females together).
Which of the following is true regarding the random variables X and Y?
A) Both X and Y can be well approximated by normal random variables.
B) Only X can be well approximated by a normal random variable.
C) Only Y can be well approximated by a normal random variable.
D) Neither X nor Y can be well approximated by a normal random variable.

Answer 1

Therefore, the answer is Neither X nor Y can be well approximated by a normal random variable.

Option D

Explanation:

Here X ≈ B (n = 20 , p = 0.08)

Y ≈ B (n = 40 , p = 0.01)

To use normal approximation the condition is np > 5 , nq > 5

For X,

`np=20 * 0.08\\\\=1.6\\\\<5`

`nq=20 * 0.98\\\\=18.4\\\\>5`

For Y

`np=40 * 0.01\\\\=0.4\\\\<5`

`nq=20*0.98\\\\=39.6\\\\>5`

So, in the both cases the value of np < 5 so normal approximation is not good.

Therefore, the answer is Neither X nor Y can be well approximated by a normal random variable.

Option D

Answer 2

D) Neither X nor Y can be well approximated by a normal random variable.

Explanation:

X = number of males (out of the 20) who are colorblind.

Y = number of females (out of the 40) who are colorblind.

Z = total number of colorblind individuals in the sample (males and females together).

The condition to use the normal approximation is that np > 5 and nq > 5

For X:

n = 20, p = 8% = 0.08,

q = 1 - 0.08 = 0.92

np = 20 * 0.08

np = 1.6 ( np < 5)

np = 20 * 0.92

np = 18.4 ( nq > 5)

For Y:

n = 40, p = 1% = 0.01,

q = 1 - 0.01 = 0.99

np = 40 * 0.01

np = 0.4 ( np < 5)

np = 40 * 0.99

np = 39.6 ( nq > 5)

For both X and Y, np < 5 and nq > 5. Since both np and nq are not greater than 5, both samples cannot be approximated by a normal distribution.