Exercise 3 Page 692

We want to describe how we can use two-way tables to find possible associations between two different categories from the same sample group. To answer the question, let's consider an example.

Example

The two-way table below shows the results of a survey where $100$ participants were asked if they were raised in a city and if they can swim.

Based on the results, we want to know if there is a relationship between being raised in a city and being able to swim. We can do that through finding and analyzing the relative frequencies between the categories. But first, let's find the joint frequencies. This means adding the frequencies in each row and each column. Now, we can find the relative frequencies by row or by column. To find the relative frequencies by row, we will write the ratios of each value to the total in that row. We will then calculate the quotients.

Let's take a closer look at the results. From the table we can read that $0.02$ or $2\%$ of the people who were not raised in a city cannot swim.

To decide if there is an association between being raised in a city and being able to swim, we should look at the first column of the table. Out of the people who were raised in a city, $0.54$ or $54\%$ can swim. Also, $0.98$ or $98\%$ of the respondents who did not grow up in a city can swim.

Since $0.98>0.54,$ a person who was not raised in a city is more likely to be able to swim. In other words, there is an association between being raised outside of a city and being able to swim.

Summary

If we calculate and then compare the relative frequencies in the two-way table, we can see if a member of one category is more likely to fall into the other category. In this case, we found that respondents who were raised outside a city were more likely to be able to swim.