Start by filling in the two-way table with the given values.
Relative Frequencies:
Cat
No Cat
Total
Dog
45;45/123≈ 0.37
125;125/177≈ 0.71
170
No Dog
78;78/123≈0.63
52;52/177≈0.29
130
Total
123
177
300
Example Solution: Most people who visited the store and have a cat do not have a dog. Most people who visited the store and do not have a cat have a dog.
Practice makes perfect
A two-way table is a frequency table that displays data collected from one source that belongs to two different categories. One category of data is represented by rows and the other is represented by columns. We want to construct a two-way table for the given data and interpret the relative frequencies by column. To do so, we will follow four steps.
Construct an empty table with the appropriate column and row headers.
Fill in the table with the given values.
Find the rest of values.
Find and interpret the relative frequencies by column.
Let's do these things one at a time.
Constructing the Table
We can start by looking at the given Venn diagram.
We can see that the diagram shows the types pets owned by the store's customers. Notice that the types of pets include cats and dogs. This information is enough to determine the appropriate column and row headers for our table.
Cat
No Cat
Total
Dog
No Dog
Total
Filling in the Table
To fill in the table, we can look at the diagram once again.
Notice that 78 customers own a cat and no dogs, 125 customers own a dog and no cats, and 45 customers own a cat and a dogs. We can also see that 52 customers own neither cats and nor dogs. Let's write the given information in our table.
Cat
No Cat
Total
Dog
45;
125;
No Dog
78;
52;
Total
Finding the Missing Values
Let's calculate the sum of each column and each row to find the missing values.
Own a Dog:& 45 + 125 = 170
Own No Dog:& 78 + 52 = 130
Own a Cat:& 45 + 78 = 123
Own No Cat:& 125 + 52 = 177
We can write the newly-obtained information in our table.
Cat
No Cat
Total
Dog
45;
125;
170
No Dog
78;
52;
130
Total
123
177
Finally, we have two ways of calculating the grand total. We can add the number of customers who own a dog to the number of customers who own no dogs, or we can add the customers who own a cat to the customers who own no cats. These two numbers must be the same!
Grand Total
l 170+130 = 300 123+177= 300 ✓
Now we can complete our table!
Cat
No Cat
Total
Dog
45;
125;
170
No Dog
78;
52;
130
Total
123
177
300
Finding and Interpreting the Relative Frequencies by Column
To find the relative frequencies by column, we calculate the ratios of each value in that column. We will round the results to the nearest hundredth.
Cat
No Cat
Total
Dog
45;45/123≈ 0.37
125;125/177≈ 0.71
170
No Dog
78;78/123≈ 0.63
52;52/177≈ 0.29
130
Total
123
177
300
Looking at the relative frequencies, we can say that most of the people who visited the store and have a cat do not have a dog. Most customers who do not have a cat have a dog.
