Exercise 4 Page 614

The conditional relative frequency is typically defined as the ratio of a joint relative frequency to the marginal relative frequency. However, there are actually two different ways to find the conditional relative frequencies. Let's explore both methods by considering an example of two-way table. Please note that this is only one of many possible examples that we could have used to illustrate these methods.

First Method: Using Relative Frequencies

This table shows the joint and marginal relative frequencies of a survey that asked $80$ students whether they play an instrument and whether they play a sport.

Now, let's calculate the conditional relative frequencies using the column totals.

Second Method: Using Joint and Marginal Frequencies

Another way to calculate these frequencies is by finding the ratio of the joint frequencies to the marginal frequencies. To show this, let's look at the two-way table that shows the joint and marginal frequencies of the same survey.

Let's calculate the ratios using the column totals.

We can see that the results are very similar to those found when using relative frequencies. This is because the first one uses the percent form while the second one uses the decimal form. Therefore, another way to calculate the conditional relative frequencies is using the ratio of the joint frequency to the marginal frequency. We have shown two ways to calculate conditional relative frequencies. In any given situation, the one that we would choose to use depends on how our data is presented.