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This two-way table summarizes data from a survey of 200 people who reported their home environment (urban or rural) and pet preference (dog or cat).
| urban | rural | total | |
|---|---|---|---|
| cat | 54 | 42 | 96 |
| dog | 80 | 24 | 104 |
| total | 134 | 66 | 200 |
What do you notice? What do you wonder?
Decide who will be Partner A and who will be Partner B.
The result of Partner A’s roll is represented by the values on the left side of the table. The result of Partner B’s roll is represented by the values on the top of the table.
Roll your number cube. Record the result of the roll. For example, if Partner A rolls a 3 and Partner B rolls a 5, then make a mark in the third row down and fifth column over. Repeat this process as many times as you can until your teacher tells you to stop.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | ||||||
| 2 | ||||||
| 3 | ||||||
| 4 | ||||||
| 5 | ||||||
| 6 |
A company has an office in Austin, Texas, and an office in Copenhagen, Denmark. The company wants to know how employees get to work, so they take a survey of all the employees and summarize the results in a table.
| walk | car | public transit | bike | total | |
|---|---|---|---|---|---|
| Austin | 63 | 376 | 125 | 63 | 627 |
| Copenhagen | 48 | 67 | 95 | 267 | 477 |
| total | 111 | 443 | 220 | 330 | 1,104 |
A school district is interested in how students get to school, so they survey their high school students to see how they get to school, and they separate the numbers by grade level. The results of the survey are summarized in the table.
| car | bus | other method | total | |
|---|---|---|---|---|
| grade 9 | 1,141 | 3,196 | 228 | 4,565 |
| grade 10 | 1,126 | 1,770 | 322 | 3,218 |
| grade 11 | 1,732 | 799 | 133 | 2,664 |
| grade 12 | 1,676 | 447 | 111 | 2,234 |
| total | 5,675 | 6,212 | 794 | 12,681 |
Tables provide a useful structure for organizing data. When several responses have been collected about some categorical variables, the data can be organized into a frequency table. The table can be used to calculate relative frequencies, which can be interpreted as probabilities.
For example, 243 participants in a survey responded to questions about their favorite season and whether they prefer wearing pants or shorts.
The results are summarized in the table.
| pants | shorts | |
|---|---|---|
| winter | 21 | 16 |
| spring | 43 | 20 |
| summer | 18 | 56 |
| autumn | 40 | 29 |
This table can be turned into a relative frequency table by dividing each of the values in the cells by the total number of participants.
| pants | shorts | |
|---|---|---|
| winter | 0.09 | 0.07 |
| spring | 0.18 | 0.08 |
| summer | 0.07 | 0.23 |
| autumn | 0.16 | 0.12 |
If a person is randomly selected from among these 243 participants, we can see that the probability that the chosen person’s favorite season is spring and likes shorts better than pants is 0.08. We can also use the fact that there were 63 people who listed spring as their favorite season (), so the probability that a randomly selected person from this group likes spring best is around 0.26 ().
Sometimes it is helpful to consider probabilities within subgroups. For example, we might want to focus on only people who prefer winter and find the probability that a person in the survey who enjoys winter best prefers wearing shorts. For this situation, we are looking at only the 37 people () who prefer winter, so that becomes the sample space, and the probability that a randomly selected person from that group likes wearing shorts is . We could compare that to the probability of a person who prefers shorts from among those who like summer best, .