Random Sampling

lessonbunny.com

Answer Key & Explanations

Random Sampling · Grade 7 · Worksheet 1

1. Charlotte wants to estimate how many students at her school of 1,472 students prefer spring as their favorite season. She randomly selects 92 students from the school roster and asks them. In her sample, 37 students say spring is their favorite season. Based on this random sample, approximately how many students in the entire school would you expect to prefer spring? Answer: 592 Solution: Find the proportion of students in the sample who prefer spring. 37 out of 92 students = 37/92 Simplify the fraction (optional, but helpful). 37/92 cannot be simplified (37 is prime and does not divide 92).
   Full step-by-step solution

   Step 1: Find the proportion of students in the sample who prefer spring. 37 out of 92 students = 37/92 Step 2: Simplify the fraction (optional, but helpful). 37/92 cannot be simplified (37 is prime and does not divide 92). Step 3: Apply the same proportion to the total school population. Expected number = (37/92) x 1,472 Step 4: Calculate. First, 1,472 divided by 92 = 16 (since 92 x 16 = 1,472) Then, 16 x 37 = 592 Step 5: State the answer. Approximately 592 students in the entire school would be expected to prefer spring. The answer is 592.

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. 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.

4. Mason surveys 250 randomly selected students at his school and finds that 85 of them own a pet cat. If the school has 1400 students, what is the best estimate for the number of students who own a pet cat? Answer: 476 Solution: Find the proportion of students who own a cat in the sample: 85 out of 250 = 85/250 = 17/50 = 0.34. Use this proportion to estimate the number in the whole school: 0.34 × 1400 = 476.
   Full step-by-step solution

   Step 1: Find the proportion of students who own a cat in the sample: 85 out of 250 = 85/250 = 17/50 = 0.34. Step 2: Use this proportion to estimate the number in the whole school: 0.34 × 1400 = 476. Step 3: Alternatively, set up a proportion: 85/250 = x/1400. Cross-multiply: 85 × 1400 = 250x → 119000 = 250x → x = 119000/250 = 476. The best estimate is that 476 students own a pet cat.

5. (-4)² + 3 × (-5) - 18 ÷ 6 = ? Answer: -2 Solution: Calculate the exponent: (-4)² = 16 Perform multiplication: 3 × (-5) = -15 Perform division: 18 ÷ 6 = 3 Rewrite the expression: 16 + (-15) - 3 Add 16 and -15: 16 + (-15) = 1 Subtract 3: 1 - 3 = -2 The answer is -2.
   Full step-by-step solution

   Step 1: Calculate the exponent: (-4)² = 16 Step 2: Perform multiplication: 3 × (-5) = -15 Step 3: Perform division: 18 ÷ 6 = 3 Step 4: Rewrite the expression: 16 + (-15) - 3 Step 5: Add 16 and -15: 16 + (-15) = 1 Step 6: Subtract 3: 1 - 3 = -2 The answer is -2.

6. Mere surveys 90 randomly selected students at her school and finds that 54 of them own a bicycle. If the school has 1500 students, what is the best estimate for the number of students who own a bicycle? Answer: 900 Solution: Find the proportion of students who own a bicycle in the sample: 54 out of 90 = 54/90 = 3/5 = 0.6 Apply this proportion to the total school population: 0.6 × 1500 = 900 Alternatively, set up a proportion: 54/90 = x/1500.
   Full step-by-step solution

   Step 1: Find the proportion of students who own a bicycle in the sample: 54 out of 90 = 54/90 = 3/5 = 0.6 Step 2: Apply this proportion to the total school population: 0.6 × 1500 = 900 Step 3: Alternatively, set up a proportion: 54/90 = x/1500. Cross-multiply: 54 × 1500 = 90x → 81000 = 90x → x = 81000/90 = 900 The best estimate is that 900 students own a bicycle.

7. Noah surveys 150 randomly selected students at his school and finds that 42 of them walk to school. If the school has 1200 students, estimate how many students walk to school. Answer: 336 Solution: Find the proportion of students who walk in the sample: 42 out of 150, which is 42/150 = 0.28. Multiply this proportion by the total school population: 0.28 × 1200 = 336.
   Full step-by-step solution

   Step 1: Find the proportion of students who walk in the sample: 42 out of 150, which is 42/150 = 0.28. Step 2: Multiply this proportion by the total school population: 0.28 × 1200 = 336. Step 3: So, an estimate for the number of students who walk to school is 336. The answer is 336.