a Each category in the table can be seen as an event.
B
b Each value in a total row or column represents the total frequency for one category.
C
c The values in the totals row and column are equal to the sum of the joint probabilities in any row or column.
A
a See solution.
B
b See solution.
C
c See solution.
Practice makes perfect
a The total rows and total columns of a two-way table are called marginal frequencies. These are given by the sum of the joint frequencies of their corresponding row or column, which are the values in the interior of the table. As an example, consider the given table.
Exam Score≥85 %
Exam Score<85%
Totals
Studied More Than 4 Hours
9
3
12
Studied Less Than 4 Hours
5
8
13
Totals
14
11
25
Each interior value gives the joint event that is described by its corresponding row and column. For example, the first cell tells us that 9 students studied more than 4 hours and got an exam score greater than or equal to 85 % . Because of this, the interior values are called joint frequencies.
b The values in the total row or column, excluding the bottom-right value, represent the total frequencies for each category. For example, the first entry of the total row tells how many students scored greater than or equal to 85 %.
14 students got a exam score≥85 %
This time, we are only focusing on one category. It is why these values are called marginal frequencies.
c Let's first calculate the joint and marginal relative frequencies. To do so, we need to calculate the ratio of each value to the total frequency. Then we will describe what they represent. Consider that the value of the bottom-right cell is 1 because it represents the total frequency.
Exam Score≥85 %
Exam Score<85%
Totals
Studied More Than 4 Hours
9/25=0.36
3/25=0.12
12/25=0.48
Studied Less Than 4 Hours
5/25=0.20
8/25=0.32
13/25=0.52
Totals
14/25=0.56
11/25=0.44
1
Now, note that the values in the total row or total column represent the outcomes but only focus on one category. They represent the marginal relative frequency, but in probability they are called marginal probabilities.
