Chapter 5.3, Problem 71E

To determine

Whether the events “homeowner” and “high school graduate” independent.

Answer to Problem 71E

No, the two events are not independent.

Explanation of Solution

Given information:

Data for relationship between educational achievement and home ownership in two − way table:

  

Two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

According to conditional probability,

  $P\left(\text{B}|\text{A}\right)=\frac{P\left(\text{A}\cap \text{B}\right)}{P\left(\text{A}\right)}=\frac{P\left(\text{A}\text{and}\text{B}\right)}{P\left(\text{A}\right)}$

Let

G: Graduate

H: Homeowner

Note that

The information about 500 U.S adults is provided in the table.

Thus,

The number of possible outcomes is 500.

Also note that

In the table, 310 of the 500 U.S adults are graduates.

Thus,

The number of favorable outcomes is 310.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

  $P\left(\text{G}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{310}{500}=0.62$

Now,

Note that

In the table, 340 of the 500 U.S adults are homeowners. In this case, the number of favorable outcomes is 340 and number of possible outcomes is 500.

  $P\left(\text{H}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{340}{500}$

Now,

Also note that

In the table, 221 of the 500 U.S adults are graduates and homeowners. In this case, the number of favorable outcomes is 221 and number of possible outcomes is 500.

  $P\left(\text{H}\text{and}\text{G}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{221}{500}$

Apply the conditional probability:

  $P\left(\text{G}|\text{H}\right)=\frac{P\left(\text{G}\text{and}\text{H}\right)}{P\left(\text{H}\right)}=\frac{221/500}{340/500}=\frac{221}{340}=\frac{13}{20}=0.65$

For events G and H to be independent,

  $P\left(\text{G}|\text{H}\right)=P\left(\text{G}\right)$

And

  $P\left(\text{H}|\text{G}\right)=P\left(\text{H}\right)$

But

In this case,

Note that

  $P\left(\text{G}|\text{H}\right)=0.65$

And

  $P\left(\text{G}\right)=0.62$

Both probabilities are not same.

Thus,

The two events are not independent.