Each cell of the two-way table represents a joint frequency. What do these values tell you about the relationships between certain groups and the entire sample set?
See solution.
Practice makes perfect
We are asked how can we use a two-way table to determine probabilities. This is easier if we look at an example.
Example
We asked 40 high school students if they play a musical instrument and if they paint regularly. The results are shown in a two-way table.
Plays an Instrument
Does Not Play an Instrument
Paints Regularly
14
11
Does Not Paint Regularly
8
7
Suppose we want to find the probability that a student paints regularly and does not play an instrument. Let's remember how to calculate the probability.
probability = number of successes/number of trials
Looking at the table, we can see that the joint frequency that we need is 11. We also know that the number of trials is 40. Let's find this probability.
P(paints and does not play instrument) = number of successes/number of trials
We can also use marginal frequencies to find other probabilities. Let's calculate the marginal frequencies for the table by adding the values from each row and column.
Plays an Instrument
Does Not Play an Instrument
Total
Paints Regularly
14
11
14+ 11= 25
Does Not Paint Regularly
8
7
8+ 7= 15
Total
14+ 8= 22
11+ 7= 18
40
Suppose that we want to find the probability that a student who paints also plays an instrument. Looking at the table, the joint frequency that satisfies this is 14, while the total number of students that paint is 25. Let's find the probability!
P(plays instrument | paints) = 14/25
Conclusion
Using a two-way table, we can find probabilities by dividing the appropriate joint frequency by the total or a marginal frequency.
