Population Inferences

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Population Inferences · Grade 7 · Worksheet 3

1. Charlotte wants to estimate how many of the 14,400 residents in her town support building a new skate park. She randomly selects 240 residents and asks them. Her results are shown in a bar graph: a blue bar for 'Support' reaches a height of 84, and a red bar for 'Do Not Support' reaches a height of 156. Based on this random sample, estimate the number of residents in the entire town who support the new skate park. Answer: 5040 Solution: In the random sample of 240 residents, 84 support the skate park. The fraction that supports is 84/240. Simplify the fraction.
   Full step-by-step solution

   Step 1: In the random sample of 240 residents, 84 support the skate park. The fraction that supports is 84/240. Step 2: Simplify the fraction. Divide numerator and denominator by 12: 84 ÷ 12 = 7, 240 ÷ 12 = 20. So 84/240 = 7/20. Step 3: The total town population is 14,400 residents. Step 4: Multiply the simplified fraction by the total: (7/20) × 14,400. Step 5: First, divide 14,400 by 20: 14,400 ÷ 20 = 720. Step 6: Then multiply 720 by 7: 720 × 7 = 5,040. Step 7: So, the estimated number of residents who support the new skate park is 5,040. The answer is 5040.

2. 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.

3. Zoe caught and tagged 50 fish in a lake. Two weeks later, Zoe caught a random sample of 83 fish and found that 25 of them were tagged. Estimate the total number of fish in the lake. Answer: 166 Solution: In the second sample, 25 out of 83 fish were tagged. This is 25/83 of the sample. Assume the same proportion of the whole lake is tagged.
   Full step-by-step solution

   Step 1: In the second sample, 25 out of 83 fish were tagged. This is 25/83 of the sample. Step 2: Assume the same proportion of the whole lake is tagged. So 50 tagged fish represent 25/83 of the total population. Step 3: Total fish = 50 ÷ (25/83) = 50 × 83 / 25 = 166. The estimated total number of fish in the lake is 166.

4. 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.

5. A city is planning to build a new community center and wants to understand the preferences of its residents. They survey a random sample of 500 residents and find that 68% support building the center. If the city has a total population of 40,000 residents, what is the best estimate for the total number of residents who support building the community center? Answer: 27200 Solution: We know that in a sample of 500 residents, 68% support building the center. The total population is 40,000. We want to estimate the total number of supporters in the entire population.
   Full step-by-step solution

   Step 1: Understand the problem We know that in a sample of 500 residents, 68% support building the center. The total population is 40,000. We want to estimate the total number of supporters in the entire population. Step 2: Find the number of supporters in the sample 68% of 500 means: 68% = 68/100 = 0.68 Supporters in sample = 0.68 × 500 = 340 people. Step 3: Use the sample proportion to estimate for the whole population The sample proportion of supporters is 68%. If the sample is random, we can assume the same proportion holds for the whole population. So, estimated supporters in the population = 68% of 40,000. 68% of 40,000 = 0.68 × 40,000. Step 4: Calculate 0.68 × 40,000 = 0.68 × 40 × 1000 = (0.68 × 40) × 1000 = 27.2 × 1000 = 27,200. Step 5: Conclusion The best estimate for the total number of residents who support building the community center is 27,200.

6. Kaia is a biologist studying fish in a lake. She catches a random sample of 87 fish, tags them, and releases them. Later, she catches another random sample of 190 fish and finds that 29 of them have tags. Estimate the total number of fish in the lake. Answer: 570 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 29/190 = 87/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 29/190 = 87/x. Step 3: Cross-multiply: 29 × x = 87 × 190. Step 4: x = (87 × 190) ÷ 29 = 570. The estimated total number of fish in the lake is 570.

7. 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.