Sampling & Inference

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Sampling & Inference · Grade 11 · Worksheet 1

1. Sophia is analyzing a triangular plot of land with sides measuring 17 meters, 21 meters, and 24 meters. She needs to find the area of this triangle. What is the area in square meters? Answer: 170 Solution: Calculate the semi-perimeter (s) = (17 + 21 + 24)/2 = 62/2 = 31 meters Apply Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) Substitute values: Area = sqrt(31(31-17)(31-21)(31-24)) Calculate inside: Area = sqrt(31 × 14 × 10 × 7) Multiply: 31 × 14 = 434, 434 × 10 = 4340, 4340 × 7 = 30380 Area =…
   Full step-by-step solution

   Step 1: Calculate the semi-perimeter (s) = (17 + 21 + 24)/2 = 62/2 = 31 meters Step 2: Apply Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) Step 3: Substitute values: Area = sqrt(31(31-17)(31-21)(31-24)) Step 4: Calculate inside: Area = sqrt(31 × 14 × 10 × 7) Step 5: Multiply: 31 × 14 = 434, 434 × 10 = 4340, 4340 × 7 = 30380 Step 6: Area = sqrt(30380) = sqrt(100 × 303.8) = 10 × sqrt(303.8) ≈ 10 × 17 = 170 Step 7: The exact calculation: sqrt(30380) = sqrt(28900) = 170 Therefore, the area is 170 square meters.

2. Olivia is conducting a survey about high school students' screen time habits. She stands outside the school library and asks every 5th student who enters to complete her survey. Out of 850 students at the school, she collects 85 responses. What percentage of the student population is represented in Olivia's sample? Answer: 10 Solution: Identify the total population size: 850 students Identify the sample size collected: 85 responses Calculate the percentage: (sample size ÷ total population) × 100 (85 ÷ 850) × 100 = 0.1 × 100 = 10% The answer is 10.
   Full step-by-step solution

   Step 1: Identify the total population size: 850 students Step 2: Identify the sample size collected: 85 responses Step 3: Calculate the percentage: (sample size ÷ total population) × 100 Step 4: (85 ÷ 850) × 100 = 0.1 × 100 = 10% The answer is 10.

3. Olivia is studying smartphone ownership patterns in her city. She stands outside a popular electronics store and surveys 200 people about whether they own a smartphone. She finds that 185 out of 200 people surveyed own a smartphone. If the actual smartphone ownership rate in the city is 75%, how many people in Olivia's sample would be expected to own a smartphone if her sampling method were unbiased? Answer: 150 Solution: The actual smartphone ownership rate in the city is 75%. Olivia surveyed 200 people in total.
   Full step-by-step solution

   Step 1: The actual smartphone ownership rate in the city is 75%. Step 2: Olivia surveyed 200 people in total. Step 3: To find the expected number of smartphone owners in an unbiased sample, multiply the total sample size by the actual ownership rate. Step 4: 200 × 0.75 = 150 Step 5: Therefore, if Olivia's sampling method were unbiased, we would expect 150 people in her sample to own a smartphone.

4. Mason is analyzing a triangular arrangement of dots. The first row has 2 dots, the second row has 4 dots, and the third row has 6 dots. If this pattern continues, how many total dots are there in the first 7 rows? Answer: 56 Solution: Identify the pattern: Row 1 has 2 dots, Row 2 has 4 dots, Row 3 has 6 dots. This forms an arithmetic sequence where each row increases by 2 dots.
   Full step-by-step solution

   Step 1: Identify the pattern: Row 1 has 2 dots, Row 2 has 4 dots, Row 3 has 6 dots. This forms an arithmetic sequence where each row increases by 2 dots. Step 2: The number of dots in row n is given by: a_n = 2 + (n-1)×2 = 2n Step 3: We need the sum of the first 7 rows: S_7 = (n/2)×(first term + last term) Step 4: First term = 2, Last term (7th row) = 2×7 = 14 Step 5: S_7 = (7/2)×(2 + 14) = (7/2)×16 = 7×8 = 56 Step 6: The total number of dots is 56.

5. Aroha is conducting a survey about student opinions on school start times. She stands outside the school library at 3:00 PM and asks every 5th student who leaves whether they think school should start later. Out of the 75 students she surveys, 57 say yes. The school principal claims this shows strong support for later start times, but Aroha worries about sampling bias. If there are 1,125 students total at the school, what percentage of the student population did Aroha actually survey? Answer: 6.67 Solution: Sample size = 75 students Population size = 1,125 students Calculate the percentage of the population surveyed Percentage = (Sample size / Population size) × 100 Percentage = (75 / 1,125) × 100 75 ÷ 1,125 = 0.066666...
   Full step-by-step solution

   Step 1: Identify the sample size and population size Sample size = 75 students Population size = 1,125 students Step 2: Calculate the percentage of the population surveyed Percentage = (Sample size / Population size) × 100 Percentage = (75 / 1,125) × 100 Step 3: Perform the division 75 ÷ 1,125 = 0.066666... Step 4: Convert to percentage 0.066666... × 100 = 6.6666...% Step 5: Round to two decimal places 6.6666...% rounds to 6.67% The answer is 6.67.

6. Aroha is designing a triangular garden with sides measuring 14 meters, 15 meters, and 16 meters. She wants to place decorative stones along the entire perimeter. How many meters of decorative stones does Aroha need? Answer: 45 Solution: Identify the sides of the triangle: 14 m, 15 m, and 16 m Calculate 14 + 15 + 16 14 + 15 = 29 29 + 16 = 45 Aroha needs 45 meters of decorative stones.
   Full step-by-step solution

   Step 1: Identify the sides of the triangle: 14 m, 15 m, and 16 m Step 2: The perimeter is the sum of all three sides Step 3: Calculate 14 + 15 + 16 Step 4: 14 + 15 = 29 Step 5: 29 + 16 = 45 Aroha needs 45 meters of decorative stones.