Sampling & Inference

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Sampling & Inference · Grade 11 · Worksheet 3

1. Sophia is conducting a survey about smartphone usage among teenagers in her city. She stands outside a popular electronics store on a Saturday afternoon and asks every sixth person who enters the store to complete her survey. Which statement best describes the potential sampling bias in Sophia's method? Answer: B. The sample may overrepresent teenagers who frequently visit electronics stores Solution: Sophia is conducting her survey outside an electronics store, which means she's sampling from a specific location People who visit electronics stores are more likely to be interested in technology and electronics This creates selection bias because she's not getting a random sample of all…
   Full step-by-step solution

   Step 1: Sophia is conducting her survey outside an electronics store, which means she's sampling from a specific location Step 2: People who visit electronics stores are more likely to be interested in technology and electronics Step 3: This creates selection bias because she's not getting a random sample of all teenagers in the city Step 4: Her sample will likely overrepresent teenagers who are more tech-savvy and interested in electronics Step 5: Therefore, the sample may overrepresent teenagers who frequently visit electronics stores The correct answer is The sample may overrepresent teenagers who frequently visit electronics stores.

2. Sophia is studying the sampling method for a survey about smartphone usage. She notices the survey was conducted by calling landline phones between 9 AM and 5 PM on weekdays. If the population of interest is all adults aged 18-65 in the city, what percentage of the population is likely to be systematically excluded due to this sampling method? Assume that 31% of adults in this age group do not have landlines and 46% are typically at work during these hours, and that these groups overlap by 16%. Answer: 61 Solution: Identify the percentages for each excluded group: 31% without landlines and 46% at work during survey hours. The overlap (people both without landlines and at work) is given as 16%.
   Full step-by-step solution

   Step 1: Identify the percentages for each excluded group: 31% without landlines and 46% at work during survey hours. Step 2: The overlap (people both without landlines and at work) is given as 16%. Step 3: Use the inclusion-exclusion principle: Total excluded = (Percentage without landlines) + (Percentage at work) - (Overlap percentage). Step 4: Calculate: 31 + 46 - 16 = 61. Step 5: The total percentage excluded is 61%. The answer is 61.

3. Mason is analyzing a triangular arrangement of dots. The first row has 7 dots, the second row has 9 dots, and each subsequent row has 2 more dots than the previous row. If there are 12 rows total, how many dots are in the arrangement? Answer: 216 Solution: Identify the sequence of dots per row: 7, 9, 11, 13, ... up to 12 terms. This is an arithmetic sequence with first term a1 = 7 and common difference d = 2.
   Full step-by-step solution

   Step 1: Identify the sequence of dots per row: 7, 9, 11, 13, ... up to 12 terms. Step 2: This is an arithmetic sequence with first term a1 = 7 and common difference d = 2. Step 3: Find the last term: a12 = a1 + (12-1)d = 7 + 11×2 = 7 + 22 = 29. Step 4: Use the arithmetic series sum formula: Sn = n/2 × (a1 + an) = 12/2 × (7 + 29) = 6 × 36 = 216. Step 5: The total number of dots is 216.

4. 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.

5. Emma is studying smartphone usage patterns at her high school. She stands outside the main entrance and surveys the first 75 students who arrive. If the school has 1500 students total, what percentage of the student population did Emma sample? Answer: 5 Solution: Emma sampled 75 students out of a total population of 1500 students.
   Full step-by-step solution

   Step 1: Emma sampled 75 students out of a total population of 1500 students. Step 2: To find the percentage, divide the sample size by the total population: 75 ÷ 1500 = 0.05 Step 3: Convert the decimal to a percentage by multiplying by 100: 0.05 × 100 = 5% The answer is 5.

6. 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.