Sampling & Inference

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Sampling & Inference · Grade 11 · Worksheet 3

1. Mason is conducting a survey about high school students' study habits. He randomly selects 120 students from the 1500 students at his school and finds that 78 of them use a specific study app. Based on this sample, what is the estimated number of students in the entire school who use this study app? Answer: 975 Solution: Calculate the proportion of students in the sample who use the study app: 78/120 = 0.65 Apply this proportion to the entire school population: 0.65 × 1500 = 975 The estimated number of students in the entire school who use the study app is 975.
   Full step-by-step solution

   Step 1: Calculate the proportion of students in the sample who use the study app: 78/120 = 0.65 Step 2: Apply this proportion to the entire school population: 0.65 × 1500 = 975 Step 3: The estimated number of students in the entire school who use the study app is 975.

2. Aroha is conducting a study about student opinions on school start times. She stands outside the school library and surveys every 10th student who enters. Out of 950 students at the school, she surveys 95 students. 76 of them support later start times. What percentage of Aroha's sample supported later start times? Answer: 80 Solution: Identify the number of students in the sample who supported later start times: 76 Identify the total sample size: 95 Calculate the percentage: (76 ÷ 95) × 100 76 ÷ 95 = 0.8 0.8 × 100 = 80 The answer is 80% of the sample supported later start times.
   Full step-by-step solution

   Step 1: Identify the number of students in the sample who supported later start times: 76 Step 2: Identify the total sample size: 95 Step 3: Calculate the percentage: (76 ÷ 95) × 100 Step 4: 76 ÷ 95 = 0.8 Step 5: 0.8 × 100 = 80 Step 6: The answer is 80% of the sample supported later start times.

3. Aroha is studying the arrangement of stones in a sacred Maori pattern. The pattern consists of 7 concentric circles of stones. The innermost circle has 3 stones, and each subsequent circle has 5 more stones than the previous circle. How many stones are in the outermost circle? Answer: 33 Solution: Identify the pattern - this is an arithmetic sequence where each term increases by 5 The first term (innermost circle) is 3 stones The outermost circle is the 7th term in the sequence Use the arithmetic sequence formula: a_n = a_1 + (n-1)d Substitute values: a_7 = 3 + (7-1)×5 Calculate: a_7 = 3…
   Full step-by-step solution

   Step 1: Identify the pattern - this is an arithmetic sequence where each term increases by 5 Step 2: The first term (innermost circle) is 3 stones Step 3: The outermost circle is the 7th term in the sequence Step 4: Use the arithmetic sequence formula: a_n = a_1 + (n-1)d Step 5: Substitute values: a_7 = 3 + (7-1)×5 Step 6: Calculate: a_7 = 3 + 6×5 Step 7: Calculate: a_7 = 3 + 30 Step 8: Calculate: a_7 = 33 The answer is 33 stones in the outermost circle.

4. Isabella is studying the properties of a regular octagon. She draws a regular octagon with side length 9 units and connects every third vertex to form a smaller octagon inside. If the area of the larger octagon is 392 square units, what is the area of the smaller octagon formed by connecting every third vertex? Answer: 196 Solution: When connecting every third vertex of a regular octagon, the resulting shape is a smaller regular octagon that is similar to the original. In a regular octagon, the distance between vertices separated by 3 steps forms a smaller similar octagon.
   Full step-by-step solution

   Step 1: When connecting every third vertex of a regular octagon, the resulting shape is a smaller regular octagon that is similar to the original. Step 2: For a regular octagon, connecting every third vertex creates a smaller octagon whose side length is related to the original by a specific ratio. In a regular octagon, the distance between vertices separated by 3 steps forms a smaller similar octagon. Step 3: The ratio of similarity between the two octagons can be determined geometrically. For a regular octagon, the linear scale factor between the original and the octagon formed by connecting every third vertex is 1/√2. Step 4: Since area scales with the square of the linear dimensions, the area ratio is (1/√2)² = 1/2. Step 5: Therefore, the area of the smaller octagon is 392 × (1/2) = 196 square units. The answer is 196.

5. Matiu is conducting a survey about smartphone usage among teenagers. He stands outside a movie theater showing the latest superhero film and asks every 5th person who enters to complete his survey. Out of 250 people he approaches, 180 respond. His survey finds that 85% of respondents use their smartphones for more than 4 hours daily. What is the total number of people who responded to Matiu's survey? Answer: 180 Solution: - Matiu approached 250 people - 180 people responded to the survey - 85% of respondents use smartphones more than 4 hours daily The question asks for the total number of people who responded to the survey The number of respondents is directly stated as 180 The total number of people who…
   Full step-by-step solution

   Step 1: Identify the information given in the problem - Matiu approached 250 people - 180 people responded to the survey - 85% of respondents use smartphones more than 4 hours daily Step 2: Determine what the question is asking The question asks for the total number of people who responded to the survey Step 3: Extract the relevant number The number of respondents is directly stated as 180 Step 4: Verify this answers the question The total number of people who responded to Matiu's survey is 180 The answer is 180.

6. Kaia is conducting a study about smartphone usage patterns among teenagers in her city. She stands outside a popular electronics store and surveys every 7th teenager who enters the store about their daily screen time. Which statement best describes the potential bias in Kaia's sampling method? Answer: D. The sample may overrepresent teenagers who are highly interested in technology Solution: When evaluating sampling methods, it's important to consider whether the sampling location or method might systematically exclude or overinclude certain groups.
   Full step-by-step solution

   When evaluating sampling methods, it's important to consider whether the sampling location or method might systematically exclude or overinclude certain groups. For example, if someone surveyed people at a gym about exercise habits, they'd likely get different results than if they surveyed people at a library. The key is to think about whether the sampling method gives everyone in the population an equal chance of being selected.