Statistical Bias and Sampling

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Answer Key & Explanations

Statistical Bias and Sampling · Grade 10 · Worksheet 1

1. Matiu wants to estimate the average height of all students at his school. He stands outside the gymnasium after a basketball game and asks every 5th person who leaves to measure their height. Identify the type of sampling method used, describe the potential bias, and explain how this could affect the estimate. Answer: Convenience sampling; selection bias; the estimate will likely be too high because basketball game attendees tend to be taller than the average student. Solution: Identify the sampling method. Matiu is not using a random method; he is choosing a location and time that is convenient (after a basketball game) and selecting every 5th person. This is convenience sampling.
   Full step-by-step solution

   Step 1: Identify the sampling method. Matiu is not using a random method; he is choosing a location and time that is convenient (after a basketball game) and selecting every 5th person. This is convenience sampling. Step 2: Identify the bias. The sample is not representative of all students because people who attend basketball games are more likely to be taller (e.g., players, fans interested in sports). This is selection bias. Step 3: Explain the effect on the estimate. Because the sample over-represents taller individuals, the average height calculated from this sample will be higher than the true average height of all students at the school. The estimate is therefore biased upward.

2. Matiu is a health researcher investigating physical activity levels among high school students in his city. He posts an online survey link on a popular social media platform and asks students to voluntarily complete it. A total of 240 students respond, and the results show that 85% of them report exercising for at least 60 minutes daily. Identify and explain the type of bias most likely present in this sampling method, and describe how this bias could affect the generalizability of Matiu's findings to all high school students in the city. Answer: Voluntary response bias (or self-selection bias); students who choose to respond may have stronger opinions or habits about exercise, leading to overestimation of daily physical activity levels among the general student population. Solution: Identify the sampling method. Matiu posted a survey link on social media and allowed students to decide whether to participate. This is a voluntary response sample, where individuals self-select into the study.
   Full step-by-step solution

   Step 1: Identify the sampling method. Matiu posted a survey link on social media and allowed students to decide whether to participate. This is a voluntary response sample, where individuals self-select into the study. Step 2: Identify the type of bias. Voluntary response samples are inherently biased because they attract people who have strong feelings or specific behaviors related to the topic. This is called voluntary response bias (or self-selection bias). Step 3: Explain how this bias affects the results. Students who are highly active or passionate about exercise are more likely to notice and complete the survey. Those who are inactive or indifferent may ignore it. Therefore, the 85% figure likely overestimates the true proportion of all high school students who exercise at least 60 minutes daily. Step 4: Discuss generalizability. Because the sample is not random and is biased toward physically active students, the results cannot be generalized to the entire population of high school students in the city. The findings only reflect the habits of a self-selected subset, not the whole population. Final answer: Voluntary response bias (or self-selection bias); students who choose to respond may have stronger opinions or habits about exercise, leading to overestimation of daily physical activity levels among the general student population.

3. A local park has a rectangular garden that is 28 meters long and 14 meters wide. The garden is divided into four equal rectangular sections by two paths: one path runs parallel to the length through the middle, and the other runs parallel to the width through the middle. The paths are 2 meters wide each. Mason wants to survey park visitors about their favorite flower type. He decides to stand at the entrance of the garden and ask every 5th person who enters. Is this a biased sampling method? If so, identify the type of bias and explain why it is problematic. Answer: Yes, this is biased; it is a convenience sample (or selection bias). Solution: Identify the sampling method. Mason is standing at the garden entrance and asking every 5th person who enters.
   Full step-by-step solution

   Step 1: Identify the sampling method. Mason is standing at the garden entrance and asking every 5th person who enters. This is a systematic sampling method, but the location (the garden entrance) and the time of day (not specified but likely limited) create bias. Step 2: Determine the target population. The target population is all park visitors who might have opinions about flower types. However, Mason is only sampling people who enter the garden, not people who use other parts of the park (like the playground, sports fields, or picnic areas). People who visit the garden may have different flower preferences than those who don't. Step 3: Identify the bias type. This is a convenience sample because Mason chooses a location that is easy for him but not representative of the entire park population. It also introduces selection bias because the sample systematically excludes visitors who do not enter the garden. Step 4: Explain why it's problematic. The results of Mason's survey will not accurately reflect the opinions of all park visitors. For example, people who are interested in flowers are more likely to enter the garden, so they may overrepresent certain flower preferences. This makes the survey results unreliable for making decisions about the whole park. Step 5: Conclusion. Yes, the sampling method is biased. It is a convenience sample with selection bias because it does not represent all park visitors fairly.

4. Kaia is a school board member investigating whether students at her high school have adequate access to digital learning resources at home. She posts a link to an online survey on the school's social media page and asks students to voluntarily complete it. The survey asks about internet access, device availability, and study space at home. After one week, 150 students have responded, and the results show that 92% of respondents have high-speed internet at home. Kaia presents these results to the board, claiming that most students in the school are well-equipped for digital learning. Identify and explain two distinct types of bias present in Kaia's sampling method that could affect the validity of her conclusion for the entire student population. Answer: Voluntary response bias and coverage bias Solution: Identify the first type of bias — voluntary response bias. Kaia used a voluntary response sample by posting a link on social media and asking students to opt in. This self-selection means the 150 respondents are not representative of all students, leading to biased results.
   Full step-by-step solution

   Step 1: Identify the first type of bias — voluntary response bias. Kaia used a voluntary response sample by posting a link on social media and asking students to opt in. Students who feel strongly about the topic (e.g., those with good internet access who want to show support, or those with poor access who want to complain) are more likely to respond. This self-selection means the 150 respondents are not representative of all students, leading to biased results. Step 2: Identify the second type of bias — coverage bias (also called selection bias). By posting only on the school's social media page, Kaia excludes students who do not use social media, do not follow the school page, or have limited internet access. Ironically, students without reliable internet access at home are less likely to see the survey, so their voices are systematically left out. This means the survey does not cover the entire target population. Step 3: Explain how these biases affect the conclusion. Because the sample is voluntary and excludes students without social media or internet access, the 92% figure likely overestimates the true proportion of students with high-speed internet. Kaia's conclusion that 'most students are well-equipped' is not supported by the biased data. Final answer: Voluntary response bias and coverage bias.