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Random Sampling

Grade 7 · Statistics · Worksheet 2

  1. Matiu works for the city council and wants to estimate how many of the 24,000 residents in his city support building a new skate park. He decides to survey a random sample of 300 residents. In his sample, 96 people say they support the new skate park. Based on this random sample, approximately how many residents in the entire city would be expected to support the new skate park? Answer: ______________
  2. Mason is conducting a survey to find out how many students in his middle school of 1,400 students prefer reading over other hobbies. He uses a random sampling method and selects 175 students. In his sample, 49 students say they prefer reading. Based on this random sample, approximately how many students in the entire school would you expect to prefer reading? Answer: ______________
  3. Isabella surveys 72 randomly selected students at her school and finds that 27 of them have a pet cat. If the school has 672 students, what is the best estimate for the number of students who have a pet cat? Answer: ______________
  4. Isabella surveys 72 randomly selected students from her school and finds that 27 of them have a pet cat. If the school has 672 students, what is the best estimate for the number of students who have a pet cat? Answer: ______________
  5. Matiu is helping his town's council plan a new recycling program. The town has a population of 32,000 residents. To estimate how many residents would participate in the program, the council randomly surveys 400 residents. In the survey, 280 residents say they would participate in the recycling program. Based on this random sample, approximately how many residents in the entire town would be expected to participate in the recycling program? Answer: ______________
  6. Liam is conducting a survey about favorite school subjects at his middle school. He randomly selects 120 students from the total student population of 800. In his sample, 45 students said math was their favorite subject. Based on this sample, approximately how many students in the entire school would you expect to have math as their favorite subject? Answer: ______________
  7. Liam surveys 80 randomly selected students at his school and finds that 55 of them have a pet. If the school has 900 students, what is the best estimate for the number of students who have a pet? Answer: ______________
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Answer Key & Explanations

Random Sampling · Grade 7 · Worksheet 2

  1. Matiu works for the city council and wants to estimate how many of the 24,000 residents in his city support building a new skate park. He decides to survey a random sample of 300 residents. In his sample, 96 people say they support the new skate park. Based on this random sample, approximately how many residents in the entire city would be expected to support the new skate park? Answer: 7680 Solution: Find the proportion of supporters in the sample. 96 out of 300 people support the skate park. Proportion = 96/300 = 0.32 Apply this proportion to the total population.
    Full step-by-step solution

    Step 1: Find the proportion of supporters in the sample. 96 out of 300 people support the skate park. Proportion = 96/300 = 0.32 Step 2: Apply this proportion to the total population. Total population = 24,000 Expected supporters = 0.32 × 24,000 Step 3: Calculate the result. 0.32 × 24,000 = 7,680 Step 4: State the final answer. Based on the sample, approximately 7,680 residents in the entire city would be expected to support the new skate park. The answer is 7680.

  2. Mason is conducting a survey to find out how many students in his middle school of 1,400 students prefer reading over other hobbies. He uses a random sampling method and selects 175 students. In his sample, 49 students say they prefer reading. Based on this random sample, approximately how many students in the entire school would you expect to prefer reading? Answer: 392 Solution: Find the proportion of students in the sample who prefer reading. Sample size = 175 students Number who prefer reading in sample = 49 Proportion = 49/175 Simplify the fraction.
    Full step-by-step solution

    Step 1: Find the proportion of students in the sample who prefer reading. Sample size = 175 students Number who prefer reading in sample = 49 Proportion = 49/175 Step 2: Simplify the fraction. 49/175 = 7/25 (divide numerator and denominator by 7) Step 3: Apply this proportion to the total school population. Total students = 1,400 Estimated number who prefer reading = (7/25) * 1,400 Step 4: Calculate. First, 1,400 divided by 25 = 56 Then, 56 times 7 = 392 Final answer: Approximately 392 students in the entire school would be expected to prefer reading.

  3. Isabella surveys 72 randomly selected students at her school and finds that 27 of them have a pet cat. If the school has 672 students, what is the best estimate for the number of students who have a pet cat? Answer: 252 Solution: Find the proportion of students with a pet cat in the sample: 27 out of 72 = 27/72 = 3/8 = 0.375. Apply this proportion to the total school population: 0.375 × 672 = 252.
    Full step-by-step solution

    Step 1: Find the proportion of students with a pet cat in the sample: 27 out of 72 = 27/72 = 3/8 = 0.375. Step 2: Apply this proportion to the total school population: 0.375 × 672 = 252. Step 3: Alternatively, set up a proportion: 27/72 = x/672. Cross-multiply: 27 × 672 = 72x → 18144 = 72x → x = 18144/72 = 252. The best estimate is that 252 students have a pet cat. The answer is 252.

  4. Isabella surveys 72 randomly selected students from her school and finds that 27 of them have a pet cat. If the school has 672 students, what is the best estimate for the number of students who have a pet cat? Answer: 252 Solution: Find the proportion of students with a cat in the sample: 27 out of 72 = 27/72. Simplify the fraction: divide numerator and denominator by 9 to get 3/8. As a decimal, 3/8 = 0.375.
    Full step-by-step solution

    Step 1: Find the proportion of students with a cat in the sample: 27 out of 72 = 27/72. Simplify the fraction: divide numerator and denominator by 9 to get 3/8. As a decimal, 3/8 = 0.375. Step 2: Use this proportion to estimate the number in the whole school: 0.375 × 672 = 252. Step 3: Alternatively, set up a proportion: 27/72 = x/672. Cross-multiply: 27 × 672 = 72x → 18144 = 72x → x = 18144/72 = 252. The best estimate is that 252 students have a pet cat.

  5. Matiu is helping his town's council plan a new recycling program. The town has a population of 32,000 residents. To estimate how many residents would participate in the program, the council randomly surveys 400 residents. In the survey, 280 residents say they would participate in the recycling program. Based on this random sample, approximately how many residents in the entire town would be expected to participate in the recycling program? Answer: 22400 Solution: Find the proportion of residents in the sample who would participate. 280 out of 400 residents said yes. Proportion = 280/400.
    Full step-by-step solution

    Step 1: Find the proportion of residents in the sample who would participate. 280 out of 400 residents said yes. Proportion = 280/400. Step 2: Simplify the fraction. 280/400 = 28/40 = 7/10. This means 7/10 or 70% of the sample would participate. Step 3: Apply this proportion to the total town population. Expected participants = (7/10) x 32,000. Step 4: Calculate. 32,000 / 10 = 3,200. Then 3,200 x 7 = 22,400. The answer is 22,400.

  6. Liam is conducting a survey about favorite school subjects at his middle school. He randomly selects 120 students from the total student population of 800. In his sample, 45 students said math was their favorite subject. Based on this sample, approximately how many students in the entire school would you expect to have math as their favorite subject? Answer: 300 Solution: Liam took a random sample of 120 students from a total of 800 students. In the sample, 45 students said math was their favorite subject.
    Full step-by-step solution

    Step 1: Understand the problem Liam took a random sample of 120 students from a total of 800 students. In the sample, 45 students said math was their favorite subject. We want to estimate how many students in the whole school would say math is their favorite subject. Step 2: Find the proportion in the sample The proportion of students in the sample who like math is: 45 (students who like math) divided by 120 (total in sample). So, proportion = 45/120. Step 3: Simplify the fraction 45/120 can be simplified by dividing numerator and denominator by 15: 45 ÷ 15 = 3 120 ÷ 15 = 8 So, 45/120 = 3/8. Step 4: Apply the proportion to the whole school If the sample is random, the same proportion should hold for the whole school. Total students = 800. Expected number = (proportion) × (total students) = (3/8) × 800. Step 5: Calculate (3/8) × 800 = 3 × (800/8) = 3 × 100 = 300. Step 6: Conclusion We expect about 300 students in the entire school to have math as their favorite subject.

  7. Liam surveys 80 randomly selected students at his school and finds that 55 of them have a pet. If the school has 900 students, what is the best estimate for the number of students who have a pet? Answer: 619 Solution: Find the proportion of students with a pet in the sample: 55 out of 80 = 55/80 = 11/16 = 0.6875. Apply this proportion to the total school population: 0.6875 × 900 = 618.75.
    Full step-by-step solution

    Step 1: Find the proportion of students with a pet in the sample: 55 out of 80 = 55/80 = 11/16 = 0.6875. Step 2: Apply this proportion to the total school population: 0.6875 × 900 = 618.75. Step 3: Since we are estimating a number of students, round to the nearest whole number: 619. Alternatively, set up a proportion: 55/80 = x/900. Cross-multiply: 55 × 900 = 80x → 49500 = 80x → x = 49500/80 = 618.75, which rounds to 619. The best estimate is 619 students.