Population Inferences

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Population Inferences · Grade 7 · Worksheet 2

1. A wildlife conservation team is studying the population of deer in a large forest preserve. They capture and tag 150 deer in the spring. During their summer survey, they capture 200 deer and find that 25 of them have tags. Based on this capture-recapture data, what is the estimated total deer population in the preserve? Answer: 1200 Solution: Set up the proportion using the capture-recapture method formula: (tagged in first sample)/(total population) = (tagged in second sample)/(total in second sample) Plug in the known values: 150/N = 25/200 Cross-multiply to solve for N: 150 × 200 = 25 × N Calculate: 30,000 = 25N Divide both sides…
   Full step-by-step solution

   Step 1: Set up the proportion using the capture-recapture method formula: (tagged in first sample)/(total population) = (tagged in second sample)/(total in second sample) Step 2: Plug in the known values: 150/N = 25/200 Step 3: Cross-multiply to solve for N: 150 × 200 = 25 × N Step 4: Calculate: 30,000 = 25N Step 5: Divide both sides by 25: N = 30,000 ÷ 25 Step 6: Calculate the final answer: N = 1,200 The estimated total deer population is 1,200.

2. Maya is a biologist studying fish in a lake. She catches a random sample of 80 fish, tags them, and releases them. Later, she catches another random sample of 51 fish and finds that 20 of them have tags. Estimate the total number of fish in the lake. Answer: 204 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 20/51 = 80/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 20/51 = 80/x. Step 3: Cross-multiply: 20 × x = 80 × 51. Step 4: x = (80 × 51) ÷ 20 = 204. The estimated total number of fish in the lake is 204.

3. Mere is investigating whether students at her school prefer playing basketball or netball. She surveys a random sample of 40 students. In her sample, 25 students say they prefer basketball. If there are 720 students in the whole school, about how many students would you expect to prefer basketball? Use a bar graph to support your reasoning: imagine a bar graph where the bar for basketball reaches 25 out of 40 on the vertical axis, and the bar for netball reaches 15 out of 40. Based on this visual sample data, what is the best estimate for the number of students in the entire school who prefer basketball? Answer: 450 Solution: Find the proportion of students who prefer basketball in the sample. The sample has 40 students, and 25 prefer basketball. Step 2: Simplify the fraction: 25/40 = 5/8.
   Full step-by-step solution

   Step 1: Find the proportion of students who prefer basketball in the sample. The sample has 40 students, and 25 prefer basketball. So the proportion is 25/40. Step 2: Simplify the fraction: 25/40 = 5/8. Step 3: To estimate the number of students in the whole school who prefer basketball, multiply the total school population by the sample proportion: 720 * (5/8). Step 4: Calculate: 720 / 8 = 90, then 90 * 5 = 450. So the best estimate is 450 students. The answer is 450.

4. A random sample of 216 students from a large middle school shows that 96 students prefer pizza for lunch. Estimate the number of students in the entire school of 1,026 students who would prefer pizza for lunch. Answer: 456 Solution: Find the sample proportion: 96 out of 216 students prefer pizza. As a fraction, that is 96/216. Simplify by dividing numerator and denominator by 24: 96 ÷ 24 = 4, 216 ÷ 24 = 9, so the proportion is 4/9.
   Full step-by-step solution

   Step 1: Find the sample proportion: 96 out of 216 students prefer pizza. As a fraction, that is 96/216. Simplify by dividing numerator and denominator by 24: 96 ÷ 24 = 4, 216 ÷ 24 = 9, so the proportion is 4/9. Step 2: Apply this proportion to the total school population of 1,026 students: (4/9) × 1,026. Step 3: Multiply: 1,026 ÷ 9 = 114, then 114 × 4 = 456. The estimated number of students who prefer pizza is 456.

5. Hana surveyed a random sample of 96 students at school and found that 32 of them prefer pizza for lunch. If the school has 534 students in total, how many students in the entire school would you expect to prefer pizza for lunch? Answer: 178 Solution: Find the proportion of students who prefer pizza in the sample: 32 out of 96 = 32/96. Multiply this proportion by the total number of students: (32/96) × 534. Calculate: 32 × 534 ÷ 96 = 178.
   Full step-by-step solution

   Step 1: Find the proportion of students who prefer pizza in the sample: 32 out of 96 = 32/96. Step 2: Multiply this proportion by the total number of students: (32/96) × 534. Step 3: Calculate: 32 × 534 ÷ 96 = 178. We would expect about 178 students in the entire school to prefer pizza for lunch.

6. Emma surveys a random sample of 80 students at her school. 55 of them say they prefer pizza over burgers. If the school has 1200 students, estimate how many students in the entire school prefer pizza over burgers. Answer: 825 Solution: Find the proportion of students in the sample who prefer pizza: 55 out of 80 = 55/80 = 11/16 = 0.6875. Multiply this proportion by the total school population: 0.6875 × 1200 = 825.
   Full step-by-step solution

   Step 1: Find the proportion of students in the sample who prefer pizza: 55 out of 80 = 55/80 = 11/16 = 0.6875. Step 2: Multiply this proportion by the total school population: 0.6875 × 1200 = 825. Step 3: So, we estimate that about 825 students in the entire school prefer pizza over burgers. The answer is 825.

7. Matiu surveys a random sample of 240 students at his school. He finds that 156 of them walk to school. If there are 1200 students in the whole school, estimate how many students walk to school. Answer: 780 Solution: Find the proportion of students who walk to school in the sample: 156 out of 240 = 156/240 = 13/20 = 0.65. Step 2: Apply this proportion to the total school population: 0.65 × 1200 = 780.
   Full step-by-step solution

   Step 1: Find the proportion of students who walk to school in the sample: 156 out of 240 = 156/240 = 13/20 = 0.65. Step 2: Apply this proportion to the total school population: 0.65 × 1200 = 780. So, an estimated 780 students walk to school. The answer is 780.