Population Inferences

Grade 7 · statistics · 94 practice problems · read aloud

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Population Inferences: Making Smart Guesses

What is it? 🤔

A population is everyone you want to know about. A sample is a smaller group you actually survey. Making an inference means using your sample data to make a smart guess about the whole population. This is useful because it's often impossible to ask everyone!

How to Make an Inference

  1. Identify: What is the population? What is the sample?
  2. Calculate: Find the percentage or proportion in your sample.
  3. Infer: Apply that sample percentage to the total population size.
  4. State your answer: Remember it's an estimate, not an exact number.

Examples in Action

Example 1: Favorite Lunch

Population: All 500 students at your school.
Sample: 50 random students surveyed.
Data: 20 out of 50 students prefer pizza.

Step 1: Sample percentage = 20/50 = 40%.
Step 2: Apply to population: 40% of 500 = 0.4 × 500 = 200.
Inference: We can infer that about 200 students in the whole school prefer pizza.

Example 2: Book Reading

Population: 1,200 7th graders in a district.
Sample: 100 7th graders surveyed.
Data: 65 read at least 4 books last month.

Step 1: Sample proportion = 65/100 = 65%.
Step 2: Apply to population: 65% of 1,200 = 0.65 × 1200 = 780.
Inference: We can estimate that about 780 seventh graders in the district read at least 4 books.

Common Mistakes to Avoid 🚫

Exact Answers: You are making an estimate. Use words like "about," "approximately," or "around." The answer is not exact.

Bad Samples: If a sample is not random (like only asking your friends), the inference will be unreliable. A good sample represents the whole population.

Mixing up Numbers: Be clear on which number is the sample size and which is the population size. Always use the population total for your final calculation.

Tips & Tricks

Proportion Power: The fraction (or percent) in your sample is the key. Just multiply that fraction by the total population.

Word Clues: Words like "estimate," "predict," "infer," or "likely" are signals you're doing a population inference.

Check Your Work: Your final answer should make sense. It can't be larger than the total population!

How to Practice

  • Look for surveys in news articles. Try to identify the population and sample.
  • Make up your own problems! Survey 10 classmates about a preference, then infer for your whole class.
  • Always write out the steps: 1) Find sample %, 2) Multiply by population total.

Practice problems

6 of the 94, worked through step by step — try them before opening the answer.

1 √(144 + 256) = ?

Hint: First perform the addition inside the square root, then find the square root of the result. For example, with √(9 + 16), you would first calculate 9 + 16 = 25, then √25 = 5.

Show the answer

Answer: 20

  1. Look at the problem: √(144 + 256). This means we need to find the square root of the sum of 144 and 256.
  2. First, add the numbers inside the square root. 144 + 256 = 400.
  3. Now the problem becomes √(400). We need to find the square root of 400.
  4. Recall that the square root of a number is a value that, when multiplied by itself, gives the original number. We ask: what number multiplied by itself equals 400? 20 × 20 = 400.
  5. Therefore, √(400) = 20. Final Answer: 20

2 (-5)² - 3 × (18 ÷ (-2)) = ?

Hint: Remember to follow the order of operations and pay close attention to the signs when working with negative numbers and exponents.

Show the answer

Answer: 52

  1. Evaluate the exponent first: (-5)² = 25
  2. Evaluate the expression inside the parentheses: 18 ÷ (-2) = -9
  3. Perform the multiplication: 3 × (-9) = -27
  4. Perform the subtraction: 25 - (-27) = 25 + 27 = 52

The answer is 52.

3 (-3)² + 4 × (15 - 7) ÷ 2 = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction.

Show the answer

Answer: 25

  1. Parentheses** 15 - 7 = 8 So the expression becomes: (-3)² + 4 × 8 ÷ 2 **
  2. Exponents** (-3)² = 9 So the expression becomes: 9 + 4 × 8 ÷ 2 **
  3. Multiplication and Division (left to right)** First: 4 × 8 = 32 Now we have: 9 + 32 ÷ 2 Next: 32 ÷ 2 = 16 So the expression becomes: 9 + 16 **
  4. Addition** 9 + 16 = 25 **Final Answer:** 25

Let's solve the problem step by step using the order of operations (PEMDAS/BODMAS). The expression is: (-3)² + 4 × (15 - 7) ÷ 2 **

4 (-5)² - 3 × (18 ÷ 3 + 4) = ?

Hint: Remember to follow the order of operations and pay attention to negative signs when squaring numbers.

Show the answer

Answer: -5

  1. Calculate the exponent first: (-5)² = 25
  2. Work inside the parentheses: 18 ÷ 3 = 6, then 6 + 4 = 10
  3. Multiply: 3 × 10 = 30
  4. Subtract: 25 - 30 = -5

The answer is -5.

5 (-4)³ + 2 × (18 - 6) ÷ 3 = ?

Hint: Remember to follow the order of operations and pay attention to negative signs when working with exponents.

Show the answer

Answer: -56

  1. Start with the expression: (-4)³ + 2 × (18 - 6) ÷ 3
  2. Calculate the exponent: (-4)³ = -4 × -4 × -4 = 16 × -4 = -64
  3. Calculate inside parentheses: (18 - 6) = 12
  4. Multiply and divide from left to right: 2 × 12 = 24, then 24 ÷ 3 = 8
  5. Add the results: -64 + 8 = -56

The answer is -56.

6 (-4)³ + 2 × (18 - 5) ÷ 2 = ?

Hint: Remember to follow the order of operations and pay attention to negative signs when working with exponents.

Show the answer

Answer: -51

  1. Calculate the exponent: (-4)³ = -4 × -4 × -4 = 16 × -4 = -64
  2. Calculate inside parentheses: (18 - 5) = 13
  3. Perform multiplication: 2 × 13 = 26
  4. Perform division: 26 ÷ 2 = 13
  5. Add the results: -64 + 13 = -51

The answer is -51.

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