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Population Inferences

Grade 7 · Statistics · Worksheet 3

  1. A school district is conducting a survey about student transportation methods. They randomly select 250 students from a total population of 12,500 students. The survey finds that 40% of the sampled students walk to school. Based on this sample, approximately how many students in the entire district would you expect to walk to school? Answer: ______________
  2. A random sample of 135 students from a large school shows that 81 prefer digital textbooks over print. Estimate the number of students in the entire school of 1125 who prefer digital textbooks. Answer: ______________
  3. Liam surveys a random sample of 45 students at his school about whether they prefer reading physical books or e-books. In the sample, 27 students prefer physical books. Liam's school has a total of 765 students. The bar graph below shows the sample results, with a blue bar for physical books (height 27) and a green bar for e-books (height 18). Based on this random sample, estimate how many students in the entire school prefer physical books, and explain why your estimate is reasonable. Answer: ______________
  4. A city is planning to build a new community center and wants to understand the age distribution of potential users. They survey 1,200 residents and find that 35% are children (under 18), 45% are adults (18-64), and the rest are seniors (65+). If the city's total population is 85,000 people, approximately how many seniors would you expect to find in the entire city based on this sample? Answer: ______________
  5. A wildlife conservation team is studying the population of red wolves in a national park. They capture and tag 45 wolves. Two months later, they capture 60 wolves and find that 15 of them have tags. Based on this capture-recapture data, what is the estimated total population of red wolves in the park? Answer: ______________
  6. A wildlife biologist is studying the population of gray wolves in a national park. She uses a capture-recapture method where she initially tags 85 wolves. Two months later, she captures 120 wolves and finds that 15 of them have tags. Based on this data, what is the estimated total wolf population in the park? Answer: ______________
  7. Matiu surveys a random sample of 160 students at his school. He finds that 96 of them prefer summer over winter. If the school has 1200 students, estimate how many students in the entire school prefer summer. Answer: ______________
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Answer Key & Explanations

Population Inferences · Grade 7 · Worksheet 3

  1. A school district is conducting a survey about student transportation methods. They randomly select 250 students from a total population of 12,500 students. The survey finds that 40% of the sampled students walk to school. Based on this sample, approximately how many students in the entire district would you expect to walk to school? Answer: 5000 Solution: We have a sample of 250 students from a total population of 12,500 students. In the sample, 40% walk to school. We want to estimate the number of walkers in the whole population.
    Full step-by-step solution

    Step 1: Understand the problem. We have a sample of 250 students from a total population of 12,500 students. In the sample, 40% walk to school. We want to estimate the number of walkers in the whole population. Step 2: Find the number of walkers in the sample. 40% of 250 students walk to school. 40% means 40/100 = 0.4. So, number of walkers in sample = 0.4 × 250. 0.4 × 250 = 100. So, 100 students in the sample walk to school. Step 3: Set up the proportion. The sample proportion of walkers should be about the same as the population proportion. Let W be the total number of walkers in the population. Then: Walkers in sample / Sample size = Walkers in population / Population size 100 / 250 = W / 12500 Step 4: Solve for W. 100 / 250 = W / 12500 Simplify 100/250 = 2/5. So, 2/5 = W / 12500 Multiply both sides by 12500: W = (2/5) × 12500 W = 2 × (12500 / 5) 12500 / 5 = 2500 So, W = 2 × 2500 = 5000. Step 5: Conclusion. We expect about 5000 students in the entire district to walk to school.

  2. A random sample of 135 students from a large school shows that 81 prefer digital textbooks over print. Estimate the number of students in the entire school of 1125 who prefer digital textbooks. Answer: 675 Solution: Find the proportion from the sample: 81 out of 135 students prefer digital textbooks. Proportion = 81/135 = 3/5 = 0.6. Apply this proportion to the total school population of 1125: 0.6 × 1125 = 675.
    Full step-by-step solution

    Step 1: Find the proportion from the sample: 81 out of 135 students prefer digital textbooks. Proportion = 81/135 = 3/5 = 0.6. Step 2: Apply this proportion to the total school population of 1125: 0.6 × 1125 = 675. The estimated number of students who prefer digital textbooks in the entire school is 675.

  3. Liam surveys a random sample of 45 students at his school about whether they prefer reading physical books or e-books. In the sample, 27 students prefer physical books. Liam's school has a total of 765 students. The bar graph below shows the sample results, with a blue bar for physical books (height 27) and a green bar for e-books (height 18). Based on this random sample, estimate how many students in the entire school prefer physical books, and explain why your estimate is reasonable. Answer: 459 Solution: Find the proportion of students in the sample who prefer physical books. The sample size is 45, and 27 prefer physical books. Simplify: 27/45 = 3/5 (divide numerator and denominator by 9).
    Full step-by-step solution

    Step 1: Find the proportion of students in the sample who prefer physical books. The sample size is 45, and 27 prefer physical books. So the proportion is 27/45. Simplify: 27/45 = 3/5 (divide numerator and denominator by 9). As a decimal, 3/5 = 0.6. Step 2: Apply this proportion to the entire school population. The school has 765 students. Multiply: 0.6 * 765 = 459. Step 3: Check using fractions: (3/5) * 765 = (3 * 765) / 5 = 2295 / 5 = 459. Step 4: Explain why it's reasonable: The sample was random, so it is likely representative of the whole school. Since 60% of the sample prefers physical books, we expect about 60% of the entire school to prefer physical books. The answer is 459.

  4. A city is planning to build a new community center and wants to understand the age distribution of potential users. They survey 1,200 residents and find that 35% are children (under 18), 45% are adults (18-64), and the rest are seniors (65+). If the city's total population is 85,000 people, approximately how many seniors would you expect to find in the entire city based on this sample? Answer: 17000 Solution: Find the percentage of seniors in the sample. Children = 35% Adults = 45% Seniors = 100% - (35% + 45%) = 100% - 80% = 20% Apply the sample percentage to the total population.
    Full step-by-step solution

    Step 1: Find the percentage of seniors in the sample. The survey gives: Children = 35% Adults = 45% Seniors = 100% - (35% + 45%) = 100% - 80% = 20% Step 2: Apply the sample percentage to the total population. Total city population = 85,000 Expected number of seniors = 20% of 85,000 20% means 20/100 = 0.20 Step 3: Calculate the number. 0.20 × 85,000 = 17,000 Step 4: Interpret the result. Based on the sample survey, we would expect about 17,000 seniors in the entire city. Final answer: 17000

  5. A wildlife conservation team is studying the population of red wolves in a national park. They capture and tag 45 wolves. Two months later, they capture 60 wolves and find that 15 of them have tags. Based on this capture-recapture data, what is the estimated total population of red wolves in the park? Answer: 180 Solution: Identify the known values from the problem. Number of wolves initially tagged = 45 Number of wolves in the second capture = 60 Number of tagged wolves in the second capture = 15 Set up the proportion for the capture-recapture method.
    Full step-by-step solution

    Step 1: Identify the known values from the problem. Number of wolves initially tagged = 45 Number of wolves in the second capture = 60 Number of tagged wolves in the second capture = 15 Step 2: Set up the proportion for the capture-recapture method. The proportion of tagged wolves in the second sample should equal the proportion of tagged wolves in the entire population. (Number of tagged in second sample) / (Total second sample) = (Total number tagged) / (Total population) 15 / 60 = 45 / N Where N is the total estimated population. Step 3: Simplify the fraction on the left side. 15/60 = 1/4 So the equation becomes: 1/4 = 45 / N Step 4: Solve for N by cross-multiplying. 1 * N = 4 * 45 N = 180 Step 5: State the final answer. The estimated total population of red wolves is 180.

  6. A wildlife biologist is studying the population of gray wolves in a national park. She uses a capture-recapture method where she initially tags 85 wolves. Two months later, she captures 120 wolves and finds that 15 of them have tags. Based on this data, what is the estimated total wolf population in the park? Answer: 680 Solution: Set up the proportion for capture-recapture method Tagged wolves in first capture = 85 Total wolves in second capture = 120 Tagged wolves in second capture = 15 Tagged in second sample / Total in second sample = Tagged in first sample / Total population 15/120 = 85/N 15/120 = 1/8 So 1/8 = 85/N 1…
    Full step-by-step solution

    Step 1: Set up the proportion for capture-recapture method Let N be the total population Tagged wolves in first capture = 85 Total wolves in second capture = 120 Tagged wolves in second capture = 15 Step 2: The proportion should be equal: Tagged in second sample / Total in second sample = Tagged in first sample / Total population 15/120 = 85/N Step 3: Simplify the left side 15/120 = 1/8 So 1/8 = 85/N Step 4: Cross multiply 1 × N = 8 × 85 N = 680 Step 5: The estimated total wolf population is 680 wolves.

  7. Matiu surveys a random sample of 160 students at his school. He finds that 96 of them prefer summer over winter. If the school has 1200 students, estimate how many students in the entire school prefer summer. Answer: 720 Solution: Find the proportion from the sample: 96 out of 160 prefer summer. Proportion = 96/160 = 0.6. The estimated number of students who prefer summer in the entire school is 720.
    Full step-by-step solution

    Step 1: Find the proportion from the sample: 96 out of 160 prefer summer. Proportion = 96/160 = 0.6. Step 2: Apply the proportion to the total population: 0.6 × 1200 = 720. The estimated number of students who prefer summer in the entire school is 720.