Random Sampling

Grade 7 · statistics · 100 practice problems · read aloud

🔊 Listen to this explanation

Random Sampling: Your Guide to Fair Surveys

🧐 What is Random Sampling?

Random sampling is a method for picking a small group of people (a sample) from a larger group (a population) so that everyone has an equal chance of being selected. It's like a fair lottery draw! We use it to get honest, unbiased information about a whole population without having to ask everyone, which saves a ton of time and money.

👣 How to Take a Random Sample: Step-by-Step

  1. Define your population: Who are you studying? (e.g., "All 7th graders at our school.")
  2. Get a list: Find a complete list of everyone in the population. This is your sampling frame.
  3. Assign numbers: Give every person on the list a unique number.
  4. Choose randomly: Use a random method (like a number generator or drawing names from a hat) to pick your sample.

📊 Visual Examples

Example 1: Favorite School Lunch
Population: 400 students in the 7th grade.
Goal: Survey 40 students about their favorite lunch.

  1. Get a list of all 400 students from the office.
  2. Assign each student a number from 1 to 400.
  3. Use a random number generator to pick 40 numbers.
  4. Survey the students whose numbers were picked. That's your random sample!

Example 2: Library Book Preferences
Population: All 150 members of the book club.
Goal: Ask 15 members what new books to buy.

  1. Write each member's name on a slip of paper.
  2. Put all the slips into a large box and mix them up.
  3. Without looking, draw 15 slips of paper.
  4. The names you draw are your random sample.

⚠️ Common Mistakes to Avoid

Mistake 1: Convenience Sampling - Only surveying your friends or the people sitting near you. This is not random and gives biased results.

Mistake 2: Not Having a Complete List - If you miss people in the population, they have zero chance of being picked, which isn't fair.

Mistake 3: Picking "Representative" People - The point of random sampling is to let chance do the work. Don't try to hand-pick a "balanced" group.

💡 Tips & Tricks

  • Memory Aid: Remember F.A.R. - Fair for All, Randomly chosen.
  • Tech Tool: Use a spreadsheet's =RANDBETWEEN(1,400) function to generate random numbers quickly.
  • Strategy: If you're using a hat, make sure all the paper slips are the same size and folded the same way so you can't tell them apart.

🎯 How to Practice

  • Classroom Activity: Try to get a random sample of 5 students from your class. What list will you use? How will you ensure it's random?
  • Real-World Scenario: Your principal wants to know what after-school activity is most popular. Design a plan for taking a random sample of the student body.
  • Online Games: Search for "random sampling simulations" online to see the process in action!

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 (-3)² + 2 × (-5) - 7 = ?

Hint: Remember to follow the order of operations carefully, especially when working with negative numbers and exponents.

Show the answer

Answer: -8

  1. Evaluate (-3)² Since squaring a negative number gives a positive result: (-3)² = 9
  2. Evaluate 2 × (-5) Multiplying a positive by a negative gives a negative: 2 × (-5) = -10
  3. Substitute back into the expression The expression becomes: 9 + (-10) - 7
  4. Add 9 and -10 9 + (-10) = -1
  5. Subtract 7 from -1 -1 - 7 = -8 Final answer: -8

Let's solve step-by-step:

2 (-4)² + 3 × (-5) - 18 ÷ 6 = ?

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

Show the answer

Answer: -2

  1. Calculate the exponent: (-4)² = 16
  2. Perform multiplication: 3 × (-5) = -15
  3. Perform division: 18 ÷ 6 = 3
  4. Rewrite the expression: 16 + (-15) - 3
  5. Add 16 and -15: 16 + (-15) = 1
  6. Subtract 3: 1 - 3 = -2

The answer is -2.

3 (-4)² - 3 × (15 ÷ 5 - 6) + 8 = ?

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

Show the answer

Answer: 33

  1. Calculate the exponent: (-4)² = 16
  2. Calculate inside the parentheses: 15 ÷ 5 = 3, then 3 - 6 = -3
  3. Multiply: 3 × (-3) = -9
  4. Combine all terms: 16 - (-9) + 8 = 16 + 9 + 8
  5. Add: 16 + 9 = 25, then 25 + 8 = 33

The answer is 33.

4 Noah surveys 150 randomly selected students at his school and finds that 42 of them walk to school. If the school has 1200 students, estimate how many students walk to school.

Hint: Think about how the proportion in the random sample can be used to estimate the proportion in the entire population. First find the fraction of students who walk in the sample, then apply that fraction to the total school population.

Show the answer

Answer: 336

  1. Find the proportion of students who walk in the sample: 42 out of 150, which is 42/150 = 0.28.
  2. Multiply this proportion by the total school population: 0.28 × 1200 = 336.
  3. So, an estimate for the number of students who walk to school is 336.

The answer is 336.

5 A rectangular prism has a length of 12 cm, a width of 8 cm, and a height of 15 cm. If you were to draw a net of this prism, what would be the total surface area of all the faces combined?

Hint: Consider how many pairs of identical faces a rectangular prism has and what formula relates the dimensions to the total area covering the outside.

Show the answer

Answer: 792

  1. Understand the problem. A rectangular prism has 6 faces: top, bottom, front, back, left side, right side. Each pair of opposite faces has the same area.
  2. Identify the dimensions. Length (L) = 12 cm Width (W) = 8 cm Height (H) = 15 cm
  3. Calculate the area of each pair of faces. Pair 1: Front and Back faces Front face area = height × width = 15 × 8 = 120 cm² Back face area = same as front = 120 cm² Total for front and back = 120 + 120 = 240 cm² Pair 2: Left and Right faces Left face area = height × length = 15 × 12 = 180 cm² Right face area = same as left = 180 cm² Total for left and right = 180 + 180 = 360 cm² Pair 3: Top and Bottom faces Top face area = length × width = 12 × 8 = 96 cm² Bottom face area = same as top = 96 cm² Total for top and bottom = 96 + 96 = 192 cm²
  4. Add all the areas together. Total surface area = 240 + 360 + 192 First, 240 + 360 = 600 Then, 600 + 192 = 792 cm²
  5. Conclusion. The total surface area of all faces in the net is 792 cm². Answer: 792

Let's find the total surface area of the rectangular prism step by step.

6 Mere surveys 120 randomly selected students at her school and finds that 48 of them have a pet cat. If the school has 1500 students, what is the estimated number of students who have a pet cat?

Hint: Think about how the proportion in the random sample can be used to estimate the proportion in the entire population. Set up a fraction comparing the number with cats to the sample size, then use it to find the estimated total.

Show the answer

Answer: 600

  1. Find the proportion of students with a pet cat in the sample: 48 out of 120 = 48/120 = 2/5 = 0.4.
  2. Use this proportion to estimate the number in the whole school: 0.4 × 1500 = 600.
  3. Alternatively, set up a proportion: 48/120 = x/1500. Cross-multiply: 48 × 1500 = 120x → 72000 = 120x → x = 72000/120 = 600. The estimated number of students who have a pet cat is 600.
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