Question

Sampling Eye Color Based on a study by Dr. P. Sorita Soni at Indiana University, assume that eye colors in the United States are distributed as follows: 40% brown, 35% blue, 12% green, 7% gray, 6% hazel.

a. A statistics instructor collects eye color data from her students. What is the name for this type of sample?

b. Identify one factor that might make the sample from part (a) biased and not representative of the general population of people in the United States.

c. If one person is randomly selected, what is the probability that this person will have brown or blue eyes?

d. If two people are randomly selected, what is the probability that at least one of them has brown eyes?

Short answer

a. The name of the sampling method is convenience sampling.

b. The sample will not be representative of the entire population of the United States and will be biased under the condition that the subjects consist of students belonging to a specific area (a community) where people have a single type of eye color.

c. The probability of selecting a person with brown or blue eyes is equal to 0.75.

d. The probability that at least one of the two persons selected has brown eyes is equal to 0.64.

Step-by-step solution

Given information

The percentages of people in the United States having different eye colors are provided.

Eye color	Percentages
Brown	0.4
Blue	0.35
Green	0.12
Gray	0.07
Hazel	0.06

Identify the method of sampling

a.

A sample extracted by the convenience sampling method is drawn from a portion of the population that was available the closest or most conveniently to the researcher.

Here, the instructor collected the sample data from the nearest and most convenient source, that is, her students. Thus, the sampling method is convenience sampling. The sample may not be representative of the entire population.

Describe the biasness of the sample

b.

The biasness of a sample is the condition when the sample is not a true representative of the population.

One factor that may create biasness in the sample is the sampling of subjects from a specific community, ethnic group, or geographical region such that they tend to share the same color of the eye.

For example, the sample may consist of students selected from a geographical area where people tend to have only a specific eye color. In this case, the sample would be biased and not represent the entire population of the United States.

Compute the probability using the addition rule

c.

Let A be the event of selecting a person with brown eyes

Let B be the event of selecting a person with blue eyes.

The probability of selecting a person with brown eyes is equal to:

$$\begin{gathered} P\left(A\right)=\frac{40}{100} \\ =0.40 \end{gathered}$$

The probability of selecting a person with blue eyes is equal to:

$$\begin{gathered} P\left(B\right)=\frac{35}{100} \\ =0.35 \end{gathered}$$

The addition rule states that

$P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P\left(AandB\right)$

As A and B are disjoint sets,$P\left(AandB\right)=0$

The probability of selecting a person with brown or blue eyes is given as:

$$\begin{gathered} P\left(AorB\right)=P\left(A\right)+P\left(B\right) \\ =0.40+0.35 \\ =0.75 \end{gathered}$$

Therefore, the probability of selecting a person with brown or blue eyes is equal to 0.75.

Compute the probability using the multiplication rule

d.

Let A be the event of selecting at least one brown-eyed person.

The probability of at least one occurrence of an event is one minus the probability of no occurrence.

The probability of selecting a brown-eyed person is equal to:

$$\begin{gathered} P\left(\text{getting}\text{a}\text{brown - eyed}\right)=\frac{40}{100} \\ =0.4 \end{gathered}$$

The probability of selecting a “not brown-eyed person” is equal to:

$$\begin{gathered} P\left(\text{none}\text{brown - eyed}\text{person}\right)=0.6\times 0.6 \\ =0.36 \end{gathered}$$

Using the multiplication rule for two joint occurrences, if two subjects are selected, the probability of getting no brown-eyed person is equal to:

$$\begin{gathered} P\left(\text{none}\text{brown - eyed}\text{person}\right)=0.6\times 0.6 \\ =0.36 \end{gathered}$$

The probability of getting at least one brown-eyed person is equal to:

$$\begin{gathered} P\left(A\right)=1-0.36 \\ =0.64 \end{gathered}$$

Therefore, the probability of getting at least one brown-eyed person is equal to 0.64.