Random Sampling

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Random Sampling · Grade 7 · Worksheet 2

1. Isabella wants to know what proportion of students at her large middle school prefer school lunches over packed lunches. The school has 1,450 students. She decides to survey 50 students who are waiting in line for the school cafeteria on Monday. In her survey, 38 students say they prefer school lunches. Is Isabella's sampling method likely to produce a representative sample of the entire student body? Explain why or why not. Answer: No, Isabella's sample is biased because she only surveyed students who were already waiting in line for the school cafeteria, which oversamples students who prefer school lunches. Solution: Identify the population. The population is all 1,450 students at the middle school. Isabella wants to know the proportion of the entire student body that prefers school lunches.
   Full step-by-step solution

   Step 1: Identify the population. The population is all 1,450 students at the middle school. Isabella wants to know the proportion of the entire student body that prefers school lunches. Step 2: Examine the sampling method. Isabella surveyed 50 students who were waiting in line for the school cafeteria. This is not a random sample because it only includes students who are already choosing to eat school lunch that day. Step 3: Consider bias. Students who always bring packed lunches would never be in the cafeteria line, so they have no chance of being included. Even students who sometimes eat school lunch but happened to bring lunch that day are excluded. This creates a selection bias. Step 4: Predict the effect. The sample likely overestimates the true proportion of students who prefer school lunches because it only surveys students who are already in line for school lunch. The result (38 out of 50, or 76%) is probably much higher than the true percentage of the entire student body. Conclusion: Isabella's sample is not representative. A better method would be to randomly select 50 students from the entire school roster so every student has an equal chance of being chosen. The answer is no, the sample is biased.

2. A city has a population of 125,000 people. A researcher wants to survey a representative sample of this population. If she uses a sampling ratio of 1:250, how many people should be included in her sample? Answer: 500 Solution: A sampling ratio of 1:250 means that for every 250 people in the population, 1 person is selected for the sample. Write the ratio as a fraction. 1:250 means 1 out of 250, which is 1/250.
   Full step-by-step solution

   Step 1: Understand the sampling ratio. A sampling ratio of 1:250 means that for every 250 people in the population, 1 person is selected for the sample. Step 2: Write the ratio as a fraction. 1:250 means 1 out of 250, which is 1/250. Step 3: Multiply the population by this fraction to find the sample size. Population = 125,000 Sample size = 125,000 × (1/250) Step 4: Perform the division first to simplify. 125,000 ÷ 250 = ? Step 5: Calculate 125,000 ÷ 250. 125,000 ÷ 250 = 125,000 ÷ (25 × 10) = (125,000 ÷ 25) ÷ 10 125,000 ÷ 25 = 5,000 5,000 ÷ 10 = 500 Alternatively: 125,000 ÷ 250 = 125,000 ÷ 250 Cancel one zero: 12,500 ÷ 25 12,500 ÷ 25 = 500 Step 6: Conclusion. The sample size is 500 people. Final answer: 500

3. Mere wants to know the most common type of tree in a large forest with 12,500 trees. She cannot count every tree, so she uses a random sampling method. She divides the forest into 50 equal-sized sections and randomly selects 5 sections. In those 5 sections, she counts a total of 240 trees, and 72 of them are rimu trees. Based on this random sample, approximately how many rimu trees are in the entire forest? Answer: 3750 Solution: Find the proportion of rimu trees in the sample. 72 rimu trees out of 240 total trees in the sample. Proportion = 72/240 = 3/10 (after dividing numerator and denominator by 24).
   Full step-by-step solution

   Step 1: Find the proportion of rimu trees in the sample. 72 rimu trees out of 240 total trees in the sample. Proportion = 72/240 = 3/10 (after dividing numerator and denominator by 24). This means 3 out of every 10 trees in the sample are rimu. Step 2: Apply this proportion to the entire forest. Total trees in forest = 12,500. Estimated rimu trees = (3/10) × 12,500. Step 3: Calculate the result. 12,500 ÷ 10 = 1,250 1,250 × 3 = 3,750 So, based on the random sample, there are approximately 3,750 rimu trees in the entire forest.

4. Noah surveys 160 randomly selected students at his school and finds that 56 of them ride a bike to school. If the school has 1400 students, what is the best estimate for the number of students who ride a bike to school? Answer: 490 Solution: Find the proportion of students who ride a bike in the sample: 56 out of 160 = 56/160 = 7/20 = 0.35. Apply this proportion to the total school population: 0.35 × 1400 = 490.
   Full step-by-step solution

   Step 1: Find the proportion of students who ride a bike in the sample: 56 out of 160 = 56/160 = 7/20 = 0.35. Step 2: Apply this proportion to the total school population: 0.35 × 1400 = 490. Step 3: Alternatively, set up a proportion: 56/160 = x/1400. Cross-multiply: 56 × 1400 = 160x → 78400 = 160x → x = 78400/160 = 490. The best estimate is that 490 students ride a bike to school.

5. A rectangular prism is drawn with dimensions 15 cm by 12 cm by 8 cm. A smaller rectangular prism with dimensions 5 cm by 4 cm by 3 cm is removed from one corner. What is the volume of the remaining solid? Answer: 1380 Solution: Calculate the volume of the large rectangular prism Volume = length × width × height = 15 × 12 × 8 = 180 × 8 = 1440 cm³ Calculate the volume of the smaller rectangular prism that was removed Volume = length × width × height = 5 × 4 × 3 = 20 × 3 = 60 cm³ Subtract the smaller volume from the…
   Full step-by-step solution

   Step 1: Calculate the volume of the large rectangular prism Volume = length × width × height = 15 × 12 × 8 = 180 × 8 = 1440 cm³ Step 2: Calculate the volume of the smaller rectangular prism that was removed Volume = length × width × height = 5 × 4 × 3 = 20 × 3 = 60 cm³ Step 3: Subtract the smaller volume from the larger volume Remaining volume = 1440 - 60 = 1380 cm³ The answer is 1380.

6. Liam surveys 135 randomly selected students from his school to estimate how many students own a pet. He finds that 81 out of 135 own a pet. If the school has 945 students, what is the estimated number of students who own a pet? Answer: 567 Solution: Find the proportion of students who own a pet in the sample: 81 out of 135 = 81/135. Simplify the fraction: divide numerator and denominator by 27 → 81/135 = 3/5.
   Full step-by-step solution

   Step 1: Find the proportion of students who own a pet in the sample: 81 out of 135 = 81/135. Simplify the fraction: divide numerator and denominator by 27 → 81/135 = 3/5. Step 2: Use this proportion to estimate the number in the whole school: (3/5) × 945 = (3 × 945) / 5 = 2835 / 5 = 567. Step 3: Alternatively, set up a proportion: 81/135 = x/945. Cross-multiply: 81 × 945 = 135x → 76545 = 135x → x = 76545 / 135 = 567. The estimated number of students who own a pet is 567.