Question

Color-blindness 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.xom).Color-blindness is gender-based, with the majority of sufferers being males.Roughly 8% of white males have some form of color-blindness, 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 color-blind.Let Y be the number of females (out of the 40) who are color-blind.Let Z be the total number of color-blind 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.e. The remaining questions refer to the following information:

Answer 1

Correct option is (d): Neither X nor Y can be well-approximated by a normal random variable.

Explanation:

The sample size of males having color-blindness is, n (X) = 20.

The sample size of females having color-blindness is, n (Y) = 40.

The proportion of males that suffer from color-blindness is, P (X) = 0.08.

The proportion of females that suffer from color-blindness is, P (Y) = 0.01.

Now both the random variables X and Y follows a Binomial distribution,

`X\sim Bin(20, 0.08)\\Y\sim Bin(40, 0.01)`

A normal distribution is used to approximate the binomial distribution if the sample is large, i.e n ≥ 30 and the probability of success is very close to 0.50.

Also if np ≥ 10 and n (1 - p) ≥ 10, the binomial distribution can be approximated by the normal distribution.

For the sample of men (X):

`np=20*0.08=1.6<10\\n(1-p)=20* (1-0.08)=18.4>10`

In this case neither n > 30 nor p is close to 0.50.

And np < 10.

Thus, the random variable X cannot be approximated by the normal distribution.

For the sample of men (Y):

`np=40*0.01=0.4<10\\n(1-p)=40* (1-0.01)=39.6>10`

In this case n > 30 but p is not close to 0.50.

And np < 10.

Thus, the random variable Y cannot be approximated by the normal distribution.

Thus, both the random variables cannot be approximated by the normal distribution.

The correct option is (d).