Unit 8, Lesson 4
Tables of Relative Frequencies
Integrated Math 2
4.1
Warm-up
Notice and Wonder: Dog City
Standards Alignment
Building On
Addressing
S-ID.5
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Building Toward
S-CP.4
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
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?
4.2
Activity
Rolling into Tables
4.3
Activity
Traveling Methods
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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 - If an employee is selected at random, what is the probability that the employee works in Austin and drive a car to work?
- If an employee is selected at random, what is the probability that the employee works in Copenhagen and rides a bike to work?
- If an employee is selected at random, what is the probability that the employee takes public transit to work?
- If an employee from Copenhagen is selected at random, what is the probability that the employee rides a bike to work?
- If an employee who takes public transit to work is selected at random, what is the probability the employee works in Austin?
- Of the five questions above, how are the last two questions different from the first three?
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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 - If a high school student is selected at random, what is the probability that the student is in grade 9 and rides the bus to school?
- If a high school student is selected at random, what is the probability that the student is in grade 12?
- If a high school student is selected at random, what is the probability that the student takes a car to school?
- If a grade 10 student is selected at random, what is the probability that the student rides a bus to school?
- If a grade 12 student is selected at random, what is the probability that the student rides a bus to school?
- If a student who rides the bus to school is selected at random, what is the probability that the student is in grade 9?
Student Lesson Summary
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 (
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 (
Glossary
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