Random Sampling

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Random Sampling · Grade 7 · Worksheet 3

1. Charlotte surveys 72 randomly selected students at her school and finds that 27 of them have a pet cat. If the school has 672 students, what is the best estimate for the number of students who have a pet cat? Answer: 252 Solution: Find the proportion of students with a pet cat in the sample: 27 out of 72 = 27/72 = 3/8 = 0.375. Apply this proportion to the total school population: 0.375 × 672 = 252.
   Full step-by-step solution

   Step 1: Find the proportion of students with a pet cat in the sample: 27 out of 72 = 27/72 = 3/8 = 0.375. Step 2: Apply this proportion to the total school population: 0.375 × 672 = 252. Step 3: Alternatively, set up a proportion: 27/72 = x/672. Cross-multiply: 27 × 672 = 72x → 18144 = 72x → x = 18144/72 = 252. The best estimate is that 252 students have a pet cat.

2. Liam is helping his town's Parks Department decide how many new benches to install along the main trail. The trail is used by 13,500 residents. To get a representative sample, the department randomly selects 225 residents and asks them how often they use the trail. In the sample, 63 residents say they use the trail at least once a week. Based on this random sample, approximately how many residents in the entire town would be expected to use the trail at least once a week? Answer: 3780 Solution: Find the proportion of residents in the sample who use the trail at least once a week. 63 out of 225 residents = 63/225 Simplify the fraction.
   Full step-by-step solution

   Step 1: Find the proportion of residents in the sample who use the trail at least once a week. 63 out of 225 residents = 63/225 Step 2: Simplify the fraction. 63/225 = (63 ÷ 9) / (225 ÷ 9) = 7/25 Step 3: Apply this proportion to the entire town population. Total residents = 13,500 Expected weekly users = (7/25) × 13,500 Step 4: Calculate the result. First, 13,500 ÷ 25 = 540 Then, 540 × 7 = 3,780 Step 5: State the final answer. Based on the random sample, approximately 3,780 residents in the entire town would be expected to use the trail at least once a week. The answer is 3780.

3. Olivia wants to estimate the number of students in her school who prefer outdoor recess over indoor activities. Her school has a total of 1,575 students. She randomly selects 75 students and asks them. In her sample, 33 students say they prefer outdoor recess. Based on this random sample, approximately how many students in the entire school would be expected to prefer outdoor recess? Answer: 693 Solution: Find the proportion of students in the sample who prefer outdoor recess. Proportion = 33 out of 75 = 33/75. Simplify the fraction if possible.
   Full step-by-step solution

   Step 1: Find the proportion of students in the sample who prefer outdoor recess. Proportion = 33 out of 75 = 33/75. Step 2: Simplify the fraction if possible. 33/75 = 11/25 (divide numerator and denominator by 3). Step 3: Apply this proportion to the total school population. Total students = 1,575. Expected number = (11/25) × 1,575. Step 4: Calculate step by step. First, 1,575 ÷ 25 = 63. Then, 63 × 11 = 693. The answer is 693.

4. Tane surveys 200 randomly selected students at his school and finds that 72 of them have a pet cat. If the school has 1500 students, what is the best estimate for the number of students who have a pet cat? Answer: 540 Solution: Find the proportion of students with a pet cat in the sample: 72 out of 200 = 72/200 = 9/25 = 0.36 Apply this proportion to the total school population: 0.36 × 1500 = 540 Alternatively, set up a proportion: 72/200 = x/1500.
   Full step-by-step solution

   Step 1: Find the proportion of students with a pet cat in the sample: 72 out of 200 = 72/200 = 9/25 = 0.36 Step 2: Apply this proportion to the total school population: 0.36 × 1500 = 540 Step 3: Alternatively, set up a proportion: 72/200 = x/1500. Cross-multiply: 72 × 1500 = 200x → 108000 = 200x → x = 108000/200 = 540 The best estimate is that 540 students have a pet cat.

5. A rectangular prism is drawn on a coordinate plane with vertices at (0,0,0), (12,0,0), (12,8,0), (0,8,0), (0,0,15), (12,0,15), (12,8,15), and (0,8,15). If you were to draw a net of this prism showing all faces, what would be the total surface area of all the faces combined? Answer: 792 Solution: Identify the dimensions of the rectangular prism from the coordinates Length = 12 units (from x-coordinates 0 to 12) Width = 8 units (from y-coordinates 0 to 8) Height = 15 units (from z-coordinates 0 to 15) Front/back faces: length × height = 12 × 15 = 180 square units each Left/right faces:…
   Full step-by-step solution

   Step 1: Identify the dimensions of the rectangular prism from the coordinates Length = 12 units (from x-coordinates 0 to 12) Width = 8 units (from y-coordinates 0 to 8) Height = 15 units (from z-coordinates 0 to 15) Step 2: Calculate the area of each face type Front/back faces: length × height = 12 × 15 = 180 square units each Left/right faces: width × height = 8 × 15 = 120 square units each Top/bottom faces: length × width = 12 × 8 = 96 square units each Step 3: Calculate total surface area There are 2 of each face type: 2 front/back faces: 2 × 180 = 360 2 left/right faces: 2 × 120 = 240 2 top/bottom faces: 2 × 96 = 192 Step 4: Add all areas together 360 + 240 + 192 = 792 The total surface area is 792 square units.

6. Sophia surveys 121 randomly selected students from her school to estimate how many students prefer reading over drawing. She finds that 66 out of 121 prefer reading. If the school has 1,331 students, what is the estimated number of students who prefer reading? Answer: 726 Solution: Find the proportion of students who prefer reading in the sample: 66 out of 121 = 66/121. Simplify: divide numerator and denominator by 11 → 66 ÷ 11 = 6, 121 ÷ 11 = 11, so 66/121 = 6/11.
   Full step-by-step solution

   Step 1: Find the proportion of students who prefer reading in the sample: 66 out of 121 = 66/121. Simplify: divide numerator and denominator by 11 → 66 ÷ 11 = 6, 121 ÷ 11 = 11, so 66/121 = 6/11. Step 2: Use this proportion to estimate the number in the whole school: (6/11) × 1,331. Step 3: Calculate: 1,331 ÷ 11 = 121, then 121 × 6 = 726. The estimated number of students who prefer reading is 726.

7. Liam surveys 135 randomly selected students at his school and finds that 63 of them have a pet cat. If the school has 1125 students, what is the best estimate for the number of students who have a pet cat? Answer: 525 Solution: Find the proportion of students with a pet cat in the sample: 63 out of 135 = 63/135. Simplify the fraction by dividing numerator and denominator by 9: 63/135 = 7/15. As a decimal, 7/15 ≈ 0.4667.
   Full step-by-step solution

   Step 1: Find the proportion of students with a pet cat in the sample: 63 out of 135 = 63/135. Simplify the fraction by dividing numerator and denominator by 9: 63/135 = 7/15. As a decimal, 7/15 ≈ 0.4667. Step 2: Apply this proportion to the total school population: (7/15) × 1125 = 7 × (1125/15) = 7 × 75 = 525. Step 3: Alternatively, set up a proportion: 63/135 = x/1125. Cross-multiply: 63 × 1125 = 135x → 70875 = 135x → x = 70875/135 = 525. The best estimate is that 525 students have a pet cat.