Sampling & Inference

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Sampling & Inference · Grade 11 · Worksheet 2

1. Noah is conducting a study about smartphone usage patterns at his high school. He stands outside the school library and surveys every 6th student who enters between 3:00 PM and 4:00 PM. Out of the 121 students who visited the library during this time, he surveyed 16 students. Which statement best describes the potential bias in Noah's sampling method? Answer: D. The sample may overrepresent students who frequently use the library. Solution: When evaluating sampling methods, it's important to consider whether certain groups are systematically included or excluded.
   Full step-by-step solution

   When evaluating sampling methods, it's important to consider whether certain groups are systematically included or excluded. A sample taken only from library visitors during a specific hour might not represent the entire student population's habits. For example, if you surveyed only students in the cafeteria during lunch about exercise habits, you might miss athletes who eat elsewhere. The location and time of sampling can significantly influence which population segments are included.

2. 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.

3. Ava is studying geometric patterns and creates a sequence of nested squares. The largest square has side length 16 cm. Each subsequent square is inscribed by connecting the midpoints of the previous square's sides, creating a smaller rotated square inside. If this process continues infinitely, what is the total area of all the squares combined? Answer: 512 Solution: The first square has area = 16 × 16 = 256 cm² When connecting midpoints of a square, the new inscribed square has area exactly half of the original square This creates a geometric series: 256 + 128 + 64 + 32 + ...
   Full step-by-step solution

   Step 1: The first square has area = 16 × 16 = 256 cm² Step 2: When connecting midpoints of a square, the new inscribed square has area exactly half of the original square Step 3: This creates a geometric series: 256 + 128 + 64 + 32 + ... Step 4: The first term a = 256, common ratio r = 1/2 Step 5: Sum of infinite geometric series = a/(1-r) = 256/(1-1/2) = 256/(1/2) = 256 × 2 = 512 Step 6: The total area of all squares is 512 cm²

4. Sophia is conducting a study about smartphone usage patterns among teenagers. She stands outside a popular electronics store and surveys the first 80 teenagers who enter. Which statement best describes the potential sampling bias in Sophia's method? Answer: D. The sample may overrepresent teenagers who frequently visit electronics stores Solution: Identify the sampling method - Sophia is surveying teenagers at an electronics store Consider who is more likely to be at an electronics store - teenagers interested in technology or those who frequently shop for electronics Recognize that this location-based sampling excludes teenagers who…
   Full step-by-step solution

   Step 1: Identify the sampling method - Sophia is surveying teenagers at an electronics store Step 2: Consider who is more likely to be at an electronics store - teenagers interested in technology or those who frequently shop for electronics Step 3: Recognize that this location-based sampling excludes teenagers who rarely or never visit electronics stores Step 4: Understand that this creates selection bias by overrepresenting a specific subgroup of teenagers Step 5: Compare to the answer choices - Option B correctly identifies this overrepresentation The correct answer is The sample may overrepresent teenagers who frequently visit electronics stores.

5. For a school project, Olivia is investigating the favorite ice cream flavors of students in the entire school of 928 students. To save time, Olivia surveys 87 students from just one grade level and finds that 12 of them prefer chocolate. If Olivia uses this sample to estimate the number of students in the whole school who prefer chocolate, what would the estimate be? Answer: 128 Solution: In the sample, the proportion of students who prefer chocolate is 12 out of 87, or 12/87. Multiply this proportion by the total school population: (12/87) * 928. Calculate: (12 * 928) // 87 = 128.
   Full step-by-step solution

   Step 1: In the sample, the proportion of students who prefer chocolate is 12 out of 87, or 12/87. Step 2: Multiply this proportion by the total school population: (12/87) * 928. Step 3: Calculate: (12 * 928) // 87 = 128. The answer is 128 students.