Introduction
Populations and Samples is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with populations and samples.
What Is Populations and Samples?
Populations and Samples means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Populations and Samples
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
For Populations and Samples, a useful visual model is a quick drawing, array, table, number line, graph, or labeled diagram that shows what each number means.
Worked Examples
Example 1
Question: A principal wants to know the average number of hours students study per week at her school. Which is the BEST way to collect a representative sample?
- A. Survey only the students in her own homeroom
- B. Survey the first \(30\) students who arrive Monday morning
- C. Randomly select \(30\) students from each grade
- D. Survey only students on the honor roll
- This is stratified random sampling: dividing the population into groups (grades) and randomly selecting from each ensures all grade levels are represented proportionally.
- This avoids bias toward any one grade's study habits.
Answer: Randomly select \(30\) students from each grade
Example 2
Question: A school wants to know which sports are most popular among its students. Compare these two methods: Why is Method 2 more likely to produce an accurate estimate?
- A. Method 1: Ask the students in the first period gym class
- B. Method 2: Randomly select 30 students from a complete list of all students
- Random selection from all students ensures the sample is representative because it includes diverse students: athletes and non-athletes, different grade levels, and different interests.
- Method 1 only surveys gym class students, who are more likely to favor sports, making that sample biased and unrepresentative of all students.
Answer: Method 2 includes students with different interests and backgrounds
Example 3
Question: A clothing store manager wants to find out which style of jeans is most popular among teenagers in her city. If she only surveys customers who shop at her store on Saturday afternoons, what type of bias is present?
- A. Convenience sampling
- B. Voluntary response bias
- C. Leading question bias
- D. Non-response bias
- Surveying only customers at one store at one time is a convenience sample because it only includes people who are easy to reach, not a random selection of all teenagers in the city.
Answer: Convenience sampling
Real-World Word Problems
Problem 1
Question: A school wants to estimate the average amount of time students spend on homework per night. Which sample size is best?
- A. \(10\) students
- B. \(25\) students
- C. \(100\) students
- D. \(1000\) students
Why it works: A sample of \(100\) students balances accuracy with practicality. Samples of \(10\) and \(25\) are too small to reliably represent the whole school; \(1000\) would give similar accuracy but be costly and time-consuming to conduct.
Answer: \(100\) students
Problem 2
Question: A school cafeteria manager wants to know which vegetables students prefer. She surveys only students who buy lunch on a given day. What population is she actually not surveying?
- A. Students who eat no vegetables
- B. Students who bring lunch from home
- C. Students who buy lunch every day
- D. All students in the school
Why it works: By surveying only lunch-buyers that day, she misses students who bring lunch from home. This creates a non-response bias and an unrepresentative sample.
Answer: Students who bring lunch from home
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
A survey asks: "Don't you agree that our school lunch program provides terrible food?" What sampling problem does this question have?
- A. Convenience sample
- B. Leading question
- C. Voluntary response bias
- D. Insufficient sample size
Question 2
Which group is an example of a population in a statistical study?
- A. \(200\) students randomly selected from all high schools in Texas
- B. All teenagers who own a smartphone
- C. Shoppers interviewed at one mall on Saturday
- D. The \(50\) customers called to participate in a phone survey
Question 3
A museum director wants to estimate how many visitors will spend more than one hour at the museum. She collects data by asking every \(10\)th visitor who enters. Is this a good sampling method?
- A. No; it is a convenience sample
- B. No; it uses voluntary response bias
- C. Yes; it is a systematic random sample
- D. Yes; it is a stratified sample
Question 4
A restaurant manager posts a survey on Facebook asking customers for feedback. What bias is introduced by this sampling method?
- A. Voluntary response bias
- B. Stratified bias
- C. Systematic bias
- D. Measurement bias
Question 5
A company wants to estimate the average salary of all its employees. It randomly selects \(50\) employees and calculates the mean. The \(50\) employees are a \underline{\hspace{3cm}} of all company employees.
- A. population
- B. sample
- C. statistic
- D. parameter
Question 6
A polling company uses a phone directory to call residents and ask about voting intentions. What bias might this method have?
- A. Neither; it is a random sample
- B. Excludes people with unlisted phone numbers and cell-phone-only users
- C. Only calls during evening hours
- D. Leads respondents to lie about their voting plans
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: Leading question
The question is phrased to suggest a particular answer ("terrible food") rather than remaining neutral, which influences how people respond.
Question 2
Answer: All teenagers who own a smartphone
A population is the entire group of interest. All teenagers who own a smartphone is the complete group; the other options describe samples or parts of a larger population.
Question 3
Answer: Yes; it is a systematic random sample
Asking every \(10\)th visitor is systematic random sampling. Since the pattern (every 10th) doesn't depend on characteristics of individual visitors, it provides a representative, unbiased sample of all museum visitors.
Question 4
Answer: Voluntary response bias
Only people who choose to respond to the Facebook survey are included. This typically includes people with strong opinions (positive or negative), making it non-representative.
Question 5
Answer: sample
A sample is a subset of the population selected for study. The \(50\) employees are part of the entire employee population, so they form the sample.
Question 6
Answer: Excludes people with unlisted phone numbers and cell-phone-only users
Phone directories only include people with listed landline numbers, which excludes people with unlisted numbers and younger voters who use only cell phones. This coverage bias makes the sample unrepresentative of all voters in the area.
Connection to Standards
This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Populations and Samples becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
