Random Sampling

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Random Sampling · Grade 7 · Worksheet 3

1. Aroha is helping her town's Parks Department estimate how many of the town's 21,000 residents would use a new hiking trail if it were built. They randomly survey 375 residents and find that 105 of them say they would use the trail regularly. Based on this random sample, approximately how many residents in the entire town would be expected to use the new hiking trail? Answer: 5880 Solution: Find the proportion of the sample that would use the trail. 105 out of 375 residents said yes. Proportion = 105 / 375.
   Full step-by-step solution

   Step 1: Find the proportion of the sample that would use the trail. 105 out of 375 residents said yes. Proportion = 105 / 375. Step 2: Simplify the fraction. Divide numerator and denominator by 15: 105/15 = 7, 375/15 = 25. So the proportion is 7/25. Step 3: Apply this proportion to the total population. Expected users = (7/25) × 21,000. Step 4: Calculate. First, 21,000 ÷ 25 = 840. Then, 840 × 7 = 5,880. The answer is 5880.

2. Emma is conducting a survey about favorite music genres at her middle school. She randomly selects 150 students from the total enrollment of 1,200 students. In her sample, 36 students said rock music was their favorite genre. Based on this random sample, approximately how many students in the entire school would you expect to have rock music as their favorite genre? Answer: 288 Solution: Calculate the proportion of students who prefer rock music in the sample 36 out of 150 students prefer rock music Proportion = 36/150 = 0.24 Apply this proportion to the entire school population Total students = 1,200 Expected rock music fans = 0.24 × 1,200 0.24 × 1,200 = 288 The answer is 288.
   Full step-by-step solution

   Step 1: Calculate the proportion of students who prefer rock music in the sample 36 out of 150 students prefer rock music Proportion = 36/150 = 0.24 Step 2: Apply this proportion to the entire school population Total students = 1,200 Expected rock music fans = 0.24 × 1,200 Step 3: Calculate the result 0.24 × 1,200 = 288 The answer is 288.

3. Mason is conducting a survey to find out how many students in his middle school of 1,400 students prefer reading over other hobbies. He uses a random sampling method and selects 175 students. In his sample, 49 students say they prefer reading. Based on this random sample, approximately how many students in the entire school would you expect to prefer reading? Answer: 392 Solution: Find the proportion of students in the sample who prefer reading. Sample size = 175 students Number who prefer reading in sample = 49 Proportion = 49/175 Simplify the fraction.
   Full step-by-step solution

   Step 1: Find the proportion of students in the sample who prefer reading. Sample size = 175 students Number who prefer reading in sample = 49 Proportion = 49/175 Step 2: Simplify the fraction. 49/175 = 7/25 (divide numerator and denominator by 7) Step 3: Apply this proportion to the total school population. Total students = 1,400 Estimated number who prefer reading = (7/25) * 1,400 Step 4: Calculate. First, 1,400 divided by 25 = 56 Then, 56 times 7 = 392 Final answer: Approximately 392 students in the entire school would be expected to prefer reading.

4. Emma surveys 75 randomly selected students at her school and finds that 30 of them walk to school. If the school has 600 students, what is the best estimate for the number of students who walk to school? Answer: 240 Solution: Find the proportion of students who walk to school in the sample: 30 out of 75 = 30/75 = 2/5 = 0.4 Apply this proportion to the total school population: 0.4 × 600 = 240 The best estimate is that 240 students walk to school.
   Full step-by-step solution

   Step 1: Find the proportion of students who walk to school in the sample: 30 out of 75 = 30/75 = 2/5 = 0.4 Step 2: Apply this proportion to the total school population: 0.4 × 600 = 240 The best estimate is that 240 students walk to school. The answer is 240.

5. Sophia wants to know the most popular fruit among the 1,200 students at her middle school. She creates a survey and considers three methods for choosing participants. Method A: She stands outside the gym after a basketball game and surveys the first 80 students who leave. Method B: She puts every student's name into a computer program that randomly selects 80 names. Method C: She asks her 80 closest friends to take the survey. Which method(s) would produce a representative sample of the entire school population? Explain your reasoning. Answer: Only Method B Solution: Understand that a representative sample must reflect the entire population of 1,200 students. Every student should have an equal chance of being chosen. Evaluate Method A.
   Full step-by-step solution

   Step 1: Understand that a representative sample must reflect the entire population of 1,200 students. Every student should have an equal chance of being chosen. Step 2: Evaluate Method A. Sophia surveys only students leaving a basketball game. Students who attend basketball games may have different fruit preferences than those who do not. This method is biased because it excludes non-attendees. Step 3: Evaluate Method B. Using a computer program to randomly select 80 names from the complete list of all 1,200 students gives each student an equal chance. This is a simple random sample and is unbiased. Step 4: Evaluate Method C. Asking only her 80 closest friends is a convenience sample. Her friends likely share similar tastes or backgrounds, so their preferences may not represent the whole school. This method is biased. Step 5: Conclusion. Only Method B produces a representative sample. The answer is Only Method B.

6. Mason surveys 200 randomly selected students at his school and finds that 64 of them ride a bike to school. If the school has 1250 students, what is the best estimate for the number of students who ride a bike to school? Answer: 400 Solution: Find the proportion of students who ride a bike in the sample: 64 out of 200 = 64/200 = 8/25 = 0.32. Apply this proportion to the total school population: 0.32 × 1250 = 400.
   Full step-by-step solution

   Step 1: Find the proportion of students who ride a bike in the sample: 64 out of 200 = 64/200 = 8/25 = 0.32. Step 2: Apply this proportion to the total school population: 0.32 × 1250 = 400. Step 3: Alternatively, set up a proportion: 64/200 = x/1250. Cross-multiply: 64 × 1250 = 200x → 80000 = 200x → x = 80000/200 = 400. The best estimate is that 400 students ride a bike to school.

7. Matiu surveys 120 randomly selected students at his school and finds that 48 of them have a pet cat. If the school has 1500 students, what is the best estimate for the total 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. Apply this proportion to the total school population: 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: Apply this proportion to the total school population: 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 best estimate is that 600 students have a pet cat.