8.3.2: Larger Populations
Lesson
Let's compare larger groups.
Exercise \(\PageIndex{1}\): First Name versus Last Name
Consider the question: In general, do the students at this school have more letters in their first name or last name? How many more letters?
- What are some ways you might get some data to answer the question?
- The other day, we compared the heights of people on different teams and the lengths of songs on different albums. What makes this question about first and last names harder to answer than those questions?
Exercise \(\PageIndex{2}\): John Jacobjingleheimerschmidt
Continue to consider the question from the warm-up: In general, do the students at this school have more letters in their first name or last name? How many more letters?
- How many letters are in your first name? In your last name?
- Do the number of letters in your own first and last names give you enough information to make conclusions about students' names in your entire school? Explain your reasoning.
-
Your teacher will provide you with data from the class. Record the mean number of letters as well as the mean absolute deviation for each data set.
- The first names of the students in your class.
- The last names of the students in your class.
- Which mean is larger? By how much? What does this difference tell you about the situation?
- Do the mean numbers of letters in the first and last names for everyone in your class give you enough information to make conclusions about students’ names in your entire school? Explain your reasoning.
Exercise \(\PageIndex{3}\): Siblings and Pets
Consider the question: Do people who are the only child have more pets ?
- Earlier, we used information about the people in your class to answer a question about the entire school. Would surveying only the people in your class give you enough information to answer this new question? Explain your reasoning.
- If you had to have an answer to this question by the end of class today, how would you gather data to answer the question?
- If you could come back tomorrow with your answer to this question, how would you gather data to answer the question?
- If someone else in the class came back tomorrow with an answer that was different than yours, what would that mean? How would you determine which answer was better?
Exercise \(\PageIndex{4}\): Sampling the Population
For each question, identify the population and a possible sample .
- What is the mean number of pages for novels that were on the best seller list in the 1990s?
- What fraction of new cars sold between August 2010 and October 2016 were built in the United States?
- What is the median income for teachers in North America?
- What is the average lifespan of Tasmanian devils?
Are you ready for more?
Political parties often use samples to poll people about important issues. One common method is to call people and ask their opinions. In most places, though, they are not allowed to call cell phones. Explain how this restriction might lead to inaccurate samples of the population.
Summary
A population is a set of people or things that we want to study. Here are some examples of populations:
- All people in the world
- All seventh graders at a school
- All apples grown in the U.S.
A sample is a subset of a population. Here are some examples of samples from the listed populations:
- The leaders of each country
- The seventh graders who are in band
- The apples in the school cafeteria
When we want to know more about a population but it is not feasible to collect data from everyone in the population, we often collect data from a sample. In the lessons that follow, we will learn more about how to pick a sample that can help answer questions about the entire population.
Glossary Entries
Definition: Mean
The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.
To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. \(7+9+12+13+14=55\) and \(55\div 5=11\).
Definition: Mean Absolute Deviation (MAD)
The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11.
To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are.
\(4+2+1+2+3=12\) and \(12\div 5=2.4\)
Definition: Median
The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order.
For the data set 7, 9, 12, 13, 14, the median is 12.
For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. \(6+8=14\) and \(14\div 2=7\).
Definition: Population
A population is a set of people or things that we want to study.
For example, if we want to study the heights of people on different sports teams, the population would be all the people on the teams.
Definition: Sample
A sample is part of a population. For example, a population could be all the seventh grade students at one school. One sample of that population is all the seventh grade students who are in band.
Practice
Exercise \(\PageIndex{5}\)
Suppose you are interested in learning about how much time seventh grade students at your school spend outdoors on a typical school day.
Select all the samples that are a part of the population you are interested in.
- The 20 students in a seventh grade math class.
- The first 20 students to arrive at school on a particular day.
- The seventh grade students participating in a science fair put on by the four middle schools in a school district.
- The 10 seventh graders on the school soccer team.
- The students on the school debate team.
Exercise \(\PageIndex{6}\)
For each sample given, list two possible populations they could belong to.
- Sample: The prices for apples at two stores near your house.
- Sample: The days of the week the students in your math class ordered food during the past week.
- Sample: The daily high temperatures for the capital cities of all 50 U.S. states over the past year.
Exercise \(\PageIndex{7}\)
If 6 coins are flipped, find the probability that there is at least 1 heads.
(From Unit 8.2.3)
Exercise \(\PageIndex{8}\)
A school's art club holds a bake sale on Fridays to raise money for art supplies. Here are the number of cookies they sold each week in the fall and in the spring:
| fall | 20 | 26 | 25 | 24 | 29 | 20 | 19 | 19 | 24 | 24 |
|---|---|---|---|---|---|---|---|---|---|---|
| spring | 19 | 27 | 29 | 21 | 25 | 22 | 26 | 21 | 25 | 25 |
- Find the mean number of cookies sold in the fall and in the spring.
- The MAD for the fall data is 2.8 cookies. The MAD for the spring data is 2.6 cookies. Express the difference in means as a multiple of the larger MAD.
- Based on this data, do you think that sales were generally higher in the spring than in the fall?
(From Unit 8.3.1)
Exercise \(\PageIndex{9}\)
A school is selling candles for a fundraiser. They keep 40% of the total sales as their commission, and they pay the rest to the candle company.
| price of candle | number of candles sold |
|---|---|
| small candle: $11 | \(68\) |
| medium candle: $18 | \(45\) |
| large candle: $25 | \(21\) |
How much money must the school pay to the candle company?
(From Unit 4.3.2)