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Random Sampling

Grade 7 · Statistics · Worksheet 1

  1. Noah is helping his town's parks department estimate how many residents visit the town's main park each week. The town has a population of 12,000. The department randomly surveys 400 residents and finds that 112 of them visited the main park at least once in the past week. Based on this random sample, approximately how many residents in the entire town would be expected to have visited the main park last week? Answer: ______________
  2. A city is planning to build a new park and wants to estimate how many residents would use it regularly. The city has a population of 48,000 people. A random sample of 375 residents is surveyed, and 225 of them say they would use the park regularly. Based on this sample, how many residents in the entire city would be expected to use the park regularly? Answer: ______________
  3. 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: ______________
  4. 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: ______________
  5. 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? Answer: ______________
  6. 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: ______________
  7. Hana surveys 120 randomly selected students at her school and finds that 48 of them own a pet cat. If the school has 900 students, what is the best estimate for the number of students who own a pet cat? Answer: ______________
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Answer Key & Explanations

Random Sampling · Grade 7 · Worksheet 1

  1. Noah is helping his town's parks department estimate how many residents visit the town's main park each week. The town has a population of 12,000. The department randomly surveys 400 residents and finds that 112 of them visited the main park at least once in the past week. Based on this random sample, approximately how many residents in the entire town would be expected to have visited the main park last week? Answer: 3360 Solution: Find the proportion of residents in the sample who visited the park. 112 out of 400 residents visited the park. Proportion = 112/400.
    Full step-by-step solution

    Step 1: Find the proportion of residents in the sample who visited the park. 112 out of 400 residents visited the park. Proportion = 112/400. Step 2: Simplify the fraction. Divide numerator and denominator by 16: 112/400 = 7/25. (Or divide by 4 to get 28/100 = 0.28.) Step 3: Apply this proportion to the total population of 12,000. Expected visitors = (7/25) × 12,000. Step 4: Calculate. First, 12,000 ÷ 25 = 480. Then, 480 × 7 = 3,360. Step 5: The answer is 3,360. Based on the random sample, approximately 3,360 residents in the entire town would be expected to have visited the main park last week.

  2. A city is planning to build a new park and wants to estimate how many residents would use it regularly. The city has a population of 48,000 people. A random sample of 375 residents is surveyed, and 225 of them say they would use the park regularly. Based on this sample, how many residents in the entire city would be expected to use the park regularly? Answer: 28800 Solution: We have a city population of 48,000. A sample of 375 residents is surveyed. In the sample, 225 say they would use the park regularly.
    Full step-by-step solution

    Let's go step-by-step. **Step 1: Understand the problem** We have a city population of 48,000. A sample of 375 residents is surveyed. In the sample, 225 say they would use the park regularly. We want to estimate the number of regular users in the entire city. **Step 2: Find the proportion in the sample** Proportion = (Number who say yes in sample) / (Sample size) Proportion = 225 / 375 **Step 3: Simplify the proportion** Divide numerator and denominator by 75: 225 ÷ 75 = 3 375 ÷ 75 = 5 So proportion = 3/5 **Step 4: Convert proportion to decimal** 3/5 = 0.6 This means 60% of the sample would use the park regularly. **Step 5: Apply proportion to the whole population** Expected number in city = Proportion × Total population Expected number = 0.6 × 48,000 **Step 6: Perform the multiplication** 0.6 × 48,000 = (6/10) × 48,000 = (6 × 48,000) / 10 6 × 48,000 = 288,000 288,000 / 10 = 28,800 **Step 7: Conclusion** Based on the sample, we expect 28,800 residents in the city to use the park regularly. **Final answer:** 28800

  3. 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.

  4. 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.

  5. 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? Answer: 600 Solution: Find the proportion of students with a pet cat in the sample: 48 out of 120 = 48/120 = 2/5 = 0.4. Use this proportion to estimate the number in the whole school: 0.4 × 1500 = 600.
    Full step-by-step solution

    Step 1: Find the proportion of students with a pet cat in the sample: 48 out of 120 = 48/120 = 2/5 = 0.4. Step 2: Use this proportion to estimate the number in the whole school: 0.4 × 1500 = 600. Step 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.

  6. 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.

  7. Hana surveys 120 randomly selected students at her school and finds that 48 of them own a pet cat. If the school has 900 students, what is the best estimate for the number of students who own a pet cat? Answer: 360 Solution: Find the proportion of students who own a cat in the sample: 48 out of 120 = 48/120 = 2/5 = 0.4 Apply this proportion to the total school population: 0.4 × 900 = 360 Alternatively, set up a proportion: 48/120 = x/900.
    Full step-by-step solution

    Step 1: Find the proportion of students who own a cat in the sample: 48 out of 120 = 48/120 = 2/5 = 0.4 Step 2: Apply this proportion to the total school population: 0.4 × 900 = 360 Step 3: Alternatively, set up a proportion: 48/120 = x/900. Cross-multiply: 48 × 900 = 120x → 43200 = 120x → x = 43200/120 = 360 The best estimate is that 360 students own a pet cat.