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Population Inferences

Grade 7 · Statistics · Worksheet 1

  1. Lily is a biologist studying fish in a lake. She catches a random sample of 83 fish, tags them, and releases them. Later, she catches another random sample of 120 fish and finds that 40 of them have tags. Estimate the total number of fish in the lake. Answer: ______________
  2. A random sample of 150 students from a middle school shows that 96 students prefer pizza over burgers. If the school has 1,250 students, estimate the number of students who prefer pizza. Answer: ______________
  3. A wildlife conservation team is tracking the population of monarch butterflies in a nature preserve. They tagged 120 butterflies in the spring. During their summer survey, they captured 200 butterflies and found that 30 of them had tags. Based on this capture-recapture data, what is the estimated total population of monarch butterflies in the preserve? Answer: ______________
  4. Mason surveys a random sample of 72 students at his school. 27 of them say they prefer winter over summer. If the school has 672 students, estimate how many students in the entire school prefer winter over summer. Answer: ______________
  5. (-4)³ + 2 × (18 - 5) ÷ 2 = ? Answer: ______________
  6. Emma surveys a random sample of 75 students at her school about whether they prefer hiking or biking for an outdoor trip. She finds that 45 students prefer hiking. If there are 825 students in the whole school, estimate how many students in the entire school prefer hiking. Justify your answer by showing the proportion from the sample. Answer: ______________
  7. A wildlife biologist is studying the population of gray wolves in a national park. She uses a capture-recapture method where she tags 150 wolves in the spring. During her summer survey, she captures 180 wolves and finds that 45 of them have tags. Based on this data, what is the estimated total wolf population in the park? Answer: ______________
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Answer Key & Explanations

Population Inferences · Grade 7 · Worksheet 1

  1. Lily is a biologist studying fish in a lake. She catches a random sample of 83 fish, tags them, and releases them. Later, she catches another random sample of 120 fish and finds that 40 of them have tags. Estimate the total number of fish in the lake. Answer: 249 Solution: Set up the proportion: tagged fish in sample / total fish in sample = total tagged fish / total fish in lake. Let x = total fish in lake. Then 40/120 = 83/x.
    Full step-by-step solution

    Step 1: Set up the proportion: tagged fish in sample / total fish in sample = total tagged fish / total fish in lake. Step 2: Let x = total fish in lake. Then 40/120 = 83/x. Step 3: Cross-multiply: 40 × x = 83 × 120. Step 4: x = (83 × 120) ÷ 40 = 249. The estimated total number of fish in the lake is 249.

  2. A random sample of 150 students from a middle school shows that 96 students prefer pizza over burgers. If the school has 1,250 students, estimate the number of students who prefer pizza. Answer: 800 Solution: Find the proportion of students who prefer pizza in the sample: 96 out of 150 = 96/150 = 0.64. Step 2: Multiply the proportion by the total school population: 0.64 × 1,250 = 800.
    Full step-by-step solution

    Step 1: Find the proportion of students who prefer pizza in the sample: 96 out of 150 = 96/150 = 0.64. Step 2: Multiply the proportion by the total school population: 0.64 × 1,250 = 800. So, an estimated 800 students in the school prefer pizza.

  3. A wildlife conservation team is tracking the population of monarch butterflies in a nature preserve. They tagged 120 butterflies in the spring. During their summer survey, they captured 200 butterflies and found that 30 of them had tags. Based on this capture-recapture data, what is the estimated total population of monarch butterflies in the preserve? Answer: 800 Solution: We are using the capture-recapture method to estimate the total population. Total population = (Number tagged initially * Total captured in second sample) / Number of tagged individuals in second sample Identify the numbers from the problem.
    Full step-by-step solution

    We are using the capture-recapture method to estimate the total population. The formula for this method is: Total population = (Number tagged initially * Total captured in second sample) / Number of tagged individuals in second sample Step 1: Identify the numbers from the problem. - Initially tagged: 120 - Second sample total captured: 200 - Tagged in second sample: 30 Step 2: Plug into the formula. Total population = (120 * 200) / 30 Step 3: Multiply 120 by 200. 120 * 200 = 24000 Step 4: Divide 24000 by 30. 24000 / 30 = 800 Step 5: Interpret the result. The estimated total population of monarch butterflies in the preserve is 800. This works because the proportion of tagged butterflies in the second sample (30 out of 200) should equal the proportion of tagged butterflies in the whole population (120 out of total population).

  4. Mason surveys a random sample of 72 students at his school. 27 of them say they prefer winter over summer. If the school has 672 students, estimate how many students in the entire school prefer winter over summer. Answer: 252 Solution: Find the proportion from the sample: 27 out of 72 prefer winter. Proportion = 27/72 = 3/8 = 0.375. Apply this proportion to the total population: 0.375 × 672 = 252.
    Full step-by-step solution

    Step 1: Find the proportion from the sample: 27 out of 72 prefer winter. Proportion = 27/72 = 3/8 = 0.375. Step 2: Apply this proportion to the total population: 0.375 × 672 = 252. The estimated number of students in the entire school who prefer winter is 252.

  5. (-4)³ + 2 × (18 - 5) ÷ 2 = ? Answer: -51 Solution: Calculate the exponent: (-4)³ = -4 × -4 × -4 = 16 × -4 = -64 Calculate inside parentheses: (18 - 5) = 13 Perform multiplication: 2 × 13 = 26 Perform division: 26 ÷ 2 = 13 Add the results: -64 + 13 = -51 The answer is -51.
    Full step-by-step solution

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

  6. Emma surveys a random sample of 75 students at her school about whether they prefer hiking or biking for an outdoor trip. She finds that 45 students prefer hiking. If there are 825 students in the whole school, estimate how many students in the entire school prefer hiking. Justify your answer by showing the proportion from the sample. Answer: 495 Solution: Find the proportion of students in the sample who prefer hiking. 45 out of 75 students prefer hiking, so the proportion is 45/75 = 3/5 = 0.6. Apply this proportion to the entire school population of 825 students.
    Full step-by-step solution

    Step 1: Find the proportion of students in the sample who prefer hiking. 45 out of 75 students prefer hiking, so the proportion is 45/75 = 3/5 = 0.6. Step 2: Apply this proportion to the entire school population of 825 students. Multiply: 0.6 * 825 = 495. Step 3: Justification: Since the sample is random, the proportion in the sample is a reasonable estimate for the proportion in the whole population. Therefore, approximately 495 students in the entire school prefer hiking. The answer is 495.

  7. A wildlife biologist is studying the population of gray wolves in a national park. She uses a capture-recapture method where she tags 150 wolves in the spring. During her summer survey, she captures 180 wolves and finds that 45 of them have tags. Based on this data, what is the estimated total wolf population in the park? Answer: 600 Solution: Set up the proportion using the capture-recapture formula: tagged in first sample / total population = tagged in second sample / total in second sample Plug in the known values: 150 / total population = 45 / 180 Simplify the right side: 45/180 = 1/4 So we have: 150 / total population = 1/4 Cross…
    Full step-by-step solution

    Step 1: Set up the proportion using the capture-recapture formula: tagged in first sample / total population = tagged in second sample / total in second sample Step 2: Plug in the known values: 150 / total population = 45 / 180 Step 3: Simplify the right side: 45/180 = 1/4 Step 4: So we have: 150 / total population = 1/4 Step 5: Cross multiply: 150 × 4 = total population × 1 Step 6: Calculate: 600 = total population Step 7: The estimated total wolf population is 600.