Question

Nine percent of all men cannot distinguish between the colors red and green. This is the type of color blindness that causes problems with traffic signals. If six men are randomly selected for a study of traffic signal perceptions: (a) Determine if X = the number of men that cannot distinguish between red and green is a binomial random variable (check the conditions for the binomial setting). (b) What is the distribution of X? (c) Find the probability that exactly two of the six men cannot distinguish between red and green. (d) What are the mean and standard deviation of the random variable X?

Answer 1

Answer:

(a) The random variable X is a Binomial random variable.

(b) The random variable X  follows a Binomial distribution.

(c) The probability that exactly two of the six men are color blind is 0.0833.

(d) The mean and standard deviation of the random variable X are 0.54 and 0.701 respectively.

Step-by-step explanation:

The proportion of men who cannot distinguish between the colors red and green is, p = 0.09.

The sample selected is of size, n = 6.

(a)

The random variable X is defined as the number of men that cannot distinguish between red and green.

A binomial random variable has the following properties:

• There are fixed number of trials
• There are only two outcomes of each trial: Success or Failure.
• The probability of each trial being a success is same.
• The trials are independent of each other.

In this case the number of trials of X is 6, any man either has color blindness or not, the probability of a man having color blindness is 0.09 for all men and a man having color blindness is independent of another man having the disease.

Thus, the random variable X is a Binomial random variable.

(b)

The distribution of the random variable X is Binomial distribution.

The probability function of X is:

$P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0,1,2,...$

(c)

Compute the probability that exactly two of the six men cannot distinguish between red and green as follows:

$P(X=2)={6\choose 2}(0.09)^{2}(1-0.09)^{6-2}\\=15\times 0.0081\times 0.68574961\\=0.083318577615\\\approx0.0833$

Thus, the probability that exactly two of the six men are color blind is 0.0833.

(d)

The mean and standard deviation of a Binomial distribution are:

$Mean=np\\Standard\ deviation =\sqrt{np(1-p)}$

Compute the mean and standard deviation of the random variable X as follows:

$Mean=np=6\times0.09=0.54\\Standard\ deviation =\sqrt{np(1-p)}=\sqrt{6\times0.09\times(1-0.09)}= 0.701$

Thus, the mean and standard deviation of the random variable X are 0.54 and 0.701 respectively.

Answer 2

Using the binomial distribution, we have that:

a) There is a fixed number of trials, they are independent, and have only two possible outcomes, hence, it is a binomial variable.

b) Binomial with $n = 6, p = 0.09$

c) There is a 0.0833 = 8.33% probability that exactly two of the six men cannot distinguish between red and green.

d) The mean of the random variable X is of 0.54, with a standard deviation of 0.7.

Binomial probability distribution

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.
• n is the number of trials.
• p is the probability of a success on a single trial.

Item a:

For each men, there are only two possible outcomes, either they can distinguish between the colors red and green, or they cannot. The probability of a men distinguishing these colors is independent of any other mean, which along with the fact that there is a fixed number of trials, means that the binomial distribution is used to solve this question.

Item b:

• 9% of all men cannot distinguish these colors, hence $p = 0.09$
• Six men are selected, hence $n = 6$.

Item c:

This probability is P(X = 2), hence:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 2) = C_{6,2}.(0.09)^{2}.(0.91)^{4} = 0.0833$

0.0833 = 8.33% probability that exactly two of the six men cannot distinguish between red and green.

Item d:

The mean of the binomial distribution is:

$E(X) = np$

Hence:

$E(X) = 6(0.09) = 0.54$

The standard deviation is:

$\sqrt{V(X)} = \sqrt{np(1 - p)}$

Hence:

$\sqrt{V(X)} = \sqrt{6(0.09)(0.91)} = 0.7$

The mean of the random variable X is of 0.54, with a standard deviation of 0.7.

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