Introduction
Making Inferences from Random 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 making inferences from random samples.
What Is Making Inferences from Random Samples?
Making Inferences from Random 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 Making Inferences from Random 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
Visual Model 1
Question: Which sample shows the highest proportion of left-handed individuals?
| Sample | Left-Handed | Total |
|---|---|---|
| Sample A | 12 | 80 |
| Sample B | 15 | 75 |
| Sample C | 18 | 120 |
- A. Sample A
- B. Sample B
- C. Sample C
- D. All have equal proportions
Why it works: Sample A: \(\frac{12}{80}=0.15\) or \(15\%\); Sample B: \(\frac{15}{75}=0.20\) or \(20\%\); Sample C: \(\frac{18}{120}=0.15\) or \(15\%\). Sample B is highest.
Answer: Sample B
Visual Model 2
Question: On which day is the proportion of items sold the lowest?
| Day | Items Sold | Sample Size |
|---|---|---|
| Monday | 18 | 100 |
| Tuesday | 36 | 160 |
| Wednesday | 25 | 125 |
- A. Monday
- B. Tuesday
- C. Wednesday
- D. All are equal
Why it works: Compare each day's proportion. Monday: \(\frac{18}{100}=0.18\) (\(18\%\)). Tuesday: \(\frac{36}{160}=0.225\) (\(22.5\%\)). Wednesday: \(\frac{25}{125}=0.20\) (\(20\%\)). Monday has the lowest proportion at \(18\%\).
Answer: Monday
Worked Examples
Example 1
Question: Which region has the lowest proportion of recycled items?
| Region | Recycled Items | Sample Size |
|---|---|---|
| East | 56 | 200 |
| West | 36 | 150 |
| North | 70 | 250 |
- A. East
- B. West
- C. North
- D. All have equal proportions
- East: \(\frac{56}{200}=0.28\) or \(28\%\); West: \(\frac{36}{150}=0.24\) or \(24\%\); North: \(\frac{70}{250}=0.28\) or \(28\%\).
- West is lowest.
Answer: West
Example 2
Question: Based on Sample A shown, if a factory makes \(200\) items, about how many defects are expected?
- A. \(30\)
- B. \(40\)
- C. \(50\)
- D. \(60\)
- The dot plot shows \(6\) defects out of \(20\) items.
- Proportion: \(\frac{6}{20}=\frac{x}{200} \Rightarrow \frac{3}{10}=\frac{x}{200} \Rightarrow x = 60\).
Answer: \(60\)
Example 3
Question: In a survey about a proposal, which sample shows a higher proportion of agreement?
| Sample | Agree | Disagree | Total |
|---|---|---|---|
| Morning | 45 | 15 | 60 |
| Afternoon | 54 | 18 | 72 |
- A. Morning
- B. Afternoon
- C. Both are equal
- D. Cannot be determined
- Morning: \(\frac{45}{60}=0.75\) or \(75\%\).
- Afternoon: \(\frac{54}{72}=0.75\) or \(75\%\).
- Both show equal proportions.
Answer: Both are equal
Real-World Word Problems
Problem 1
Question: In a random sample of \(50\) students, \(18\) said they prefer science. About how many students in the whole school of \(800\) are likely to prefer science?
- A. \(180\)
- B. \(288\)
- C. \(324\)
- D. \(360\)
Why it works: Set up the proportion: \(\frac{18}{50}=\frac{x}{800}\). Solve: \(x=\frac{18\times800}{50}=288\).
Answer: \(288\)
Problem 2
Question: In a random sample of \(120\) middle school students, \(45\) play a musical instrument. The school has \(840\) students. About how many are likely to play an instrument?
- A. \(270\)
- B. \(315\)
- C. \(420\)
- D. \(504\)
Why it works: \(\frac{45}{120}=\frac{x}{840}\). Reduce to \(\frac{3}{8}=\frac{x}{840} \Rightarrow x = 315\).
Answer: \(315\)
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
In a sample of \(90\) items from a factory, \(27\) were defective. If the factory produces \(2700\) items per day, about how many are expected to be defective?
- A. \(300\)
- B. \(450\)
- C. \(810\)
- D. \(1080\)
Question 2
A health clinic surveyed \(150\) patients and found that \(42\) have high blood pressure. The clinic sees \(500\) patients per month. About how many have high blood pressure?
- A. \(84\)
- B. \(112\)
- C. \(140\)
- D. \(168\)
Question 3
In a sample of \(100\) apps on a phone, \(16\) are gaming apps. A mobile device company has \(8000\) app downloads tracked. About how many are gaming apps?
- A. \(800\)
- B. \(1000\)
- C. \(1280\)
- D. \(1600\)
Question 4
Three samples show proportions of \(\frac{15}{50}\), \(\frac{12}{40}\), and \(\frac{18}{72}\) of students who prefer science. Which two samples have the same proportion?
- A. First and second
- B. Second and third
- C. First and third
- D. None have the same proportion
Question 5
A survey of \(70\) people found \(28\) prefer hiking. Which estimate of the total population preferring hiking is most reasonable if the population is \(3500\)?
- A. \(980\)
- B. \(1200\)
- C. \(1400\)
- D. \(1500\)
Question 6
In a sample of \(160\) grocery shoppers, \(64\) bought organic produce. The store has approximately \(4000\) customers per week. About how many buy organic produce?
- A. \(1000\)
- B. \(1200\)
- C. \(1600\)
- D. \(1800\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(810\)
\(\frac{27}{90}=\frac{x}{2700}\). Reduce to \(\frac{3}{10} \Rightarrow x = 810\).
Question 2
Answer: \(140\)
\(\frac{42}{150}=\frac{x}{500}\). Simplify: \(\frac{7}{25}=\frac{x}{500} \Rightarrow x = 140\).
Question 3
Answer: \(1280\)
\(\frac{16}{100}=\frac{x}{8000}\). So \(x = 0.16 \times 8000 = 1280\).
Question 4
Answer: First and second
\(\frac{15}{50}=0.30\); \(\frac{12}{40}=0.30\); \(\frac{18}{72}=0.25\). First and second both equal \(30\%\).
Question 5
Answer: \(1400\)
\(\frac{28}{70}=\frac{x}{3500}\). Reduce: \(\frac{2}{5}=\frac{x}{3500} \Rightarrow x = 1400\).
Question 6
Answer: \(1600\)
\(\frac{64}{160}=\frac{x}{4000}\). Reduce: \(\frac{2}{5}=\frac{x}{4000} \Rightarrow x = 1600\).
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
Making Inferences from Random 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.
