Statistical Bias and Sampling

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Answer Key & Explanations

Statistical Bias and Sampling · Grade 10 · Worksheet 3

1. A triangular prism has a right triangle base with legs measuring 6 cm and 8 cm, and a height of 15 cm. The prism is sliced by a plane that passes through the midpoint of the prism's height and is parallel to the triangular bases. What is the area of the cross-section formed by this slice? Answer: 24 cm² Solution: We have a triangular prism with a right triangle base (legs 6 cm and 8 cm) and prism height 15 cm.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a triangular prism with a right triangle base (legs 6 cm and 8 cm) and prism height 15 cm. A plane cuts the prism parallel to the triangular base, at the midpoint of the prism's height (so 15/2 = 7.5 cm from one base). Because the plane is parallel to the triangular base, the cross-section is a triangle similar to the base triangle. --- **Step 2: Find the area of the base triangle** Base triangle: right triangle with legs 6 cm and 8 cm. Area of base triangle = (1/2) * (6 cm) * (8 cm) = (1/2) * 48 = 24 cm². --- **Step 3: Determine the scaling factor for the cross-section** The cross-section is parallel to the base and at the midpoint along the prism's height. But wait — in a prism, cross-sections parallel to the base are congruent to the base, not scaled, unless it's a pyramid. Let's check carefully. Actually, in a prism, all cross-sections parallel to the base are identical in shape and size to the base. So the cross-section here should be exactly the same triangle as the base triangle, regardless of where you slice parallel to the base. That means the cross-section area = area of base triangle = 24 cm². --- **Step 4: Verify the "midpoint of prism's height" detail** The problem says: "passes through the midpoint of the prism's height and is parallel to the triangular bases." Yes — in a prism, any plane parallel to the base yields a cross-section congruent to the base. So the midpoint information is irrelevant for the area — it's still the same size triangle. --- **Step 5: Conclusion** Area of cross-section = area of base triangle = 24 cm². --- **Final answer:** 24 cm²

2. A university is studying the average study time per week for undergraduate students. They randomly select 500 students from their enrollment database, but only 180 students respond to the survey. The respondents have an average study time of 18 hours per week, while the non-respondents are later found to average only 8 hours per week. Calculate the actual average study time for all 500 selected students, accounting for non-response bias. Answer: 11.6 Solution: Identify the number of respondents and non-respondents Total students selected: 500 Respondents: 180 Non-respondents: 500 - 180 = 320 Respondents average: 18 hours/week Total respondent hours = 180 × 18 = 3,240 hours Calculate the total study hours for non-respondents Non-respondents average: 8…
   Full step-by-step solution

   Step 1: Identify the number of respondents and non-respondents Total students selected: 500 Respondents: 180 Non-respondents: 500 - 180 = 320 Step 2: Calculate the total study hours for respondents Respondents average: 18 hours/week Total respondent hours = 180 × 18 = 3,240 hours Step 3: Calculate the total study hours for non-respondents Non-respondents average: 8 hours/week Total non-respondent hours = 320 × 8 = 2,560 hours Step 4: Calculate the total study hours for all 500 students Total hours = 3,240 + 2,560 = 5,800 hours Step 5: Calculate the actual average study time Average = Total hours ÷ Total students = 5,800 ÷ 500 = 11.6 hours/week The answer is 11.6.

3. Olivia wants to estimate the average number of hours students at her school spend on homework per week. She posts a survey link on the school's social media page and receives responses from 75 students. Identify the sampling method used, the type of bias present, and explain why this method may lead to an inaccurate estimate. Answer: Voluntary response sample; nonresponse bias and voluntary response bias; the sample is not representative because it only includes students who choose to respond, likely those with strong opinions or more free time. Solution: Identify the sampling method. Olivia posts a survey link on social media and waits for students to respond. This is a voluntary response sample because students choose whether to participate.
   Full step-by-step solution

   Step 1: Identify the sampling method. Olivia posts a survey link on social media and waits for students to respond. This is a voluntary response sample because students choose whether to participate. Step 2: Identify the bias type. Voluntary response samples suffer from voluntary response bias, where individuals with strong opinions (e.g., those who spend many hours or very few hours on homework) are more likely to respond. Additionally, nonresponse bias occurs because many students may not see the post or choose not to respond. Step 3: Explain why it leads to inaccurate estimates. The sample is not representative of the entire school population. For example, students who spend little time on homework may ignore the survey, while those who spend a lot may be more motivated to respond. This skews the average away from the true population mean. Final answer: Voluntary response sample; voluntary response bias and nonresponse bias; the sample is not representative, leading to an inaccurate estimate of the average homework hours.

4. Matiu wants to estimate the average height of students at his school. He surveys 150 students from the basketball team. Identify the type of sampling bias present and explain why it leads to an inaccurate estimate. Answer: Selection bias (undercoverage bias); the sample is not representative of the entire student population because basketball players tend to be taller than average, so the estimate will be too high. Solution: Identify the sampling method. Matiu uses a convenience sample by only surveying basketball team members, not randomly selecting from all students. Recognize the bias type.
   Full step-by-step solution

   Step 1: Identify the sampling method. Matiu uses a convenience sample by only surveying basketball team members, not randomly selecting from all students. Step 2: Recognize the bias type. This is selection bias, specifically undercoverage bias, because certain groups (non-athletes, shorter students) are systematically excluded. Step 3: Explain the effect. Basketball players are generally taller than the average student. Therefore, the sample's average height will be higher than the true school average, leading to an overestimate. Step 4: Conclusion. The estimate is biased and not reliable for the whole school.

5. Tane is a student council member at a large high school with 1,500 students. He wants to estimate the proportion of students who support extending the lunch period by 15 minutes. To save time, Tane stands outside the school cafeteria during the lunch period and surveys every 5th student who enters. He collects responses from 75 students, and 63 of them say they support the extension. Identify and explain the most significant type of sampling bias present in Tane's method, and discuss how this bias could affect the accuracy of his estimate for the entire student population. Answer: Convenience bias (also known as selection bias due to location and time); the sample only includes students who visit the cafeteria during lunch, systematically excluding those who bring lunch, eat elsewhere, or do not eat during that time, which may overestimate support because students who take the time to go to the cafeteria might be more likely to want a longer lunch. Solution: Identify the sampling method. Tane stands outside the cafeteria during lunch and surveys every 5th student who enters.
   Full step-by-step solution

   Step 1: Identify the sampling method. Tane stands outside the cafeteria during lunch and surveys every 5th student who enters. This is a systematic sampling method applied only to students who visit the cafeteria during that specific time period. Step 2: Identify the bias. The most significant bias is convenience bias (a form of selection bias). By sampling only at the cafeteria entrance during lunch, Tane systematically excludes: - Students who bring their own lunch and eat elsewhere (e.g., in classrooms, library, or outdoors) - Students who skip lunch or have other commitments (e.g., clubs, detentions) - Students who leave campus for lunch - Students who eat at different times (if the cafeteria has multiple lunch shifts) Step 3: Explain how this affects the estimate. Students who visit the cafeteria are likely to have a stronger interest in lunch-related issues, including extending the lunch period. They might be more supportive of the extension because they personally use the cafeteria and want more time. The 63 out of 75 (84%) who support the extension is almost certainly an overestimate for the entire school population. Students who do not use the cafeteria might be indifferent or opposed to the change, but their views are not captured. Step 4: Discuss generalizability. Because the sample is drawn from a non-representative subset of the student body, the results cannot be generalized to all 1,500 students. The true proportion of support is likely lower than 84%. Final answer: The most significant bias is convenience bias (selection bias due to location and time); the sample only includes cafeteria users, likely overestimating support for the lunch extension because it excludes students who do not eat in the cafeteria.

6. Charlotte wants to estimate the average number of hours students at her school spend on homework each week. She posts a survey on the school's social media page and receives 127 responses from students who chose to participate. Identify the sampling method used, the type of bias present, and explain why this bias is a problem. Answer: Voluntary response sampling; nonresponse bias (or voluntary response bias); the sample is not representative because it only includes students who chose to respond, likely those with strong opinions or more free time, leading to an overestimate or underestimate of the true average. Solution: Identify the sampling method. Charlotte posts a survey on social media and anyone can choose to respond. This is voluntary response sampling, not a random sample.
   Full step-by-step solution

   Step 1: Identify the sampling method. Charlotte posts a survey on social media and anyone can choose to respond. This is voluntary response sampling, not a random sample. Step 2: Identify the bias. Because participation is voluntary, the sample suffers from voluntary response bias (a type of nonresponse bias). Students with strong opinions (e.g., those who feel they have too much homework or very little) are more likely to respond, while average students may ignore the survey. Step 3: Explain why it is a problem. The sample is not representative of the entire student population. The estimated average hours could be too high (if mostly overworked students respond) or too low (if mostly underworked students respond), leading to an inaccurate conclusion about the true average homework time for all students at the school.