Question

Ages of “dot-com” employees. The age (in years) distribution for the employees of a highly successful “dot-com” company headquartered in Atlanta is shown in the next table. An employee is to be randomly selected from this population.

1. Can the relative frequency distribution in the table be interpreted as a probability distribution? Explain.
2. Graph the probability distribution.
3. What is the probability that the randomly selected employee is over 30 years of age? Over 40 years of age? Under 30 years of age?
4. What is the probability that the randomly selected employee will be 25 or 26 years old?

Short answer

a. Yes.

b. The graph is given below:

c. Over 30: 0.40

Over 40: 0

Under 30: 0.51

d. Yes.

Step-by-step solution

(a) Elucidation of the criteria

Any relative frequency distribution can be interpreted as a probability distribution if it follows particular criteria. The only criteria are that the summation of all the relative frequencies is equivalent to 1.

Summation of the relative frequencies

The summation of the relative frequencies is one as shown below:

$$\begin{gathered} \mathbf{\text{Summation=0.02+0.04+0.05+0.07+0.02+0.11+0.07+0.09+0.13+0.15+0.12}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=1}} \end{gathered}$$

Therefore, it can be deduced that the relative frequency distribution is the same as the probability distribution in this context.

(b) Criteria of a probability distribution

For relative frequency distribution to be interpreted as a probability distribution, the criteria are to see whether the summation of the relative frequencies is 1. The graphical representation of probability distribution can be drawn as the criteria have been fulfilled.

Graphical representation

The graph shown above depicts the probability distribution containing the ages of the employees with the associated proportion. The blue bars represent the associated proportion or probabilities of each age.

(c) Computation of P(Age>30)

$\mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{Age>30}}\mathbf{\right)}$refers to the probability of selecting an employee above 30 years, and the calculation is shown below:

role="math" localid="1653646235879" $$\begin{gathered} \mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{Age>30}}\mathbf{\right)}\mathbf{\text{=P}}\mathbf{\left(}\mathbf{\text{Age=31}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=32}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=33}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.13+0.15+0.12}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.41}} \end{gathered}$$

Computation of P(Age>40)

$\mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{Age>40}}\mathbf{\right)}$refers to the probability of selecting an employee above 40 years. There are no such employees present in the sample who are over 40 years because the maximum age is 33 years, so its probability must be 0.

Computation of  P(Age<30)

$\mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{Age<30}}\mathbf{\right)}$refers to the probability of selecting an employee below 30 years, and the calculation is shown below:

role="math" localid="1653646788111" $$\begin{gathered} \mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{Age<30}}\mathbf{\right)}\mathbf{\text{=P}}\mathbf{\left(}\mathbf{\text{Age=20}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=21}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=23}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=24}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=25}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=26}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=27}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=28}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=29}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.02+0.04+0.05+0.07+0.04+0.02+0.07+0.02+0.11+0.07}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{51} \end{gathered}$$

(d) Definition of the probability of selecting an employee

A probability of selecting an employee refers to the possibility of an employee getting selected from a sample of employees. In this case, the “dot-com” employees are taken into consideration.

Computation of P(Age=25 or 26)

$\mathbf{P}\mathbf{(}\mathbf{Age}\mathbf{=25}\mathbf{or}\mathbf{26}\mathbf{)}$refers to the probability of selecting an employee of 25 or 26 years of age, and the calculation is shown below:

$$\begin{gathered} \mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{Age=25or26}}\mathbf{\right)}\mathbf{\text{=P}}\mathbf{\left(}\mathbf{\text{Age=25}}\mathbf{\right)}\mathbf{\text{+P}}\mathbf{\left(}\mathbf{\text{Age=26}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.02+0.07}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.09}} \end{gathered}$$

Therefore, the probability of a “dot-com” employee of 25 or 26 years of age getting selected is 0.09.