Statistical Bias and Sampling

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Answer Key & Explanations

Statistical Bias and Sampling · Grade 10 · Worksheet 1

1. Aroha wants to estimate the average height of all students at her school. She stands outside the gymnasium after a basketball game and asks every 5th student who exits to measure their height. Identify the sampling method used, the type of bias present, and explain why this sample may not represent the entire school population. Answer: Convenience sampling; selection bias; the sample over-represents basketball game attendees who may be taller than average. Solution: Identify the sampling method. Aroha chooses a location (gymnasium) and time (after a basketball game) that is convenient, and selects every 5th person exiting. This is convenience sampling, not random sampling.
   Full step-by-step solution

   Step 1: Identify the sampling method. Aroha chooses a location (gymnasium) and time (after a basketball game) that is convenient, and selects every 5th person exiting. This is convenience sampling, not random sampling. Step 2: Identify the bias. The sample is not representative because basketball players and fans may be taller on average than the general student population. This is selection bias (or undercoverage bias). Step 3: Explain the issue. The sample excludes students who do not attend basketball games, such as those involved in other activities or who stay home. Therefore, the average height estimate will likely be too high and not reflect the true average height of all students at the school. Final answer: Convenience sampling; selection bias; the sample over-represents taller individuals.

2. Hana is a school board member investigating whether the new school library hours meet the needs of all students at a large high school. She places a survey on the library's front desk for one week and asks students to fill it out voluntarily as they visit the library. A total of 120 students complete the survey, and 96 of them say the hours are sufficient. However, the school has 800 students total. Identify and explain two distinct types of bias present in Hana's sampling method, and state why the results cannot be generalized to the entire student population. Answer: The two biases are voluntary response bias (only library visitors who choose to complete the survey are included) and selection bias (the sample excludes students who never visit the library, who may have different needs). The results cannot be generalized because the sample is not representative of all 800 students. Solution: Identify the sampling method. Hana places a survey on the library's front desk and asks voluntary participants to fill it out. The sample of 120 library visitors who voluntarily responded is not representative of the entire population of 800 students.
   Full step-by-step solution

   Step 1: Identify the sampling method. Hana places a survey on the library's front desk and asks voluntary participants to fill it out. This is a convenience sample combined with voluntary response sampling. Step 2: Identify the first bias — voluntary response bias. Only students who visit the library and choose to complete the survey are included. Students who are unhappy with the hours may be more motivated to respond, while those who are satisfied or indifferent may not bother. This leads to overrepresentation of extreme opinions. Step 3: Identify the second bias — selection bias (or undercoverage bias). The sample only includes students who visit the library. Students who never go to the library — perhaps because the hours don't suit them, or because they study elsewhere — are completely excluded from the sample. Their needs and opinions are not captured at all. Step 4: Explain why the results cannot be generalized. The sample of 120 library visitors who voluntarily responded is not representative of the entire population of 800 students. The opinions of students who do not use the library or who chose not to respond are missing. Therefore, the 80% approval rate (96 out of 120) likely does not reflect the true opinion of all students. Final answer: The two biases are voluntary response bias and selection bias; the results cannot be generalized to the entire student population because the sample is not representative.

3. Charlotte wants to estimate the average height of students at her school. She stands outside the gymnasium after a basketball game and surveys the first 40 students who exit. Identify the type of sampling bias present in this method and explain why it leads to a biased estimate. Answer: Convenience sampling bias; the sample is not representative of the entire student population because it only includes students who attended a basketball game, who are likely taller than average. Solution: Identify the sampling method. Charlotte is surveying the first 40 students exiting the gym after a basketball game.
   Full step-by-step solution

   Step 1: Identify the sampling method. Charlotte is surveying the first 40 students exiting the gym after a basketball game. This is a convenience sample because she is choosing a readily available group rather than randomly selecting from all students. Step 2: Identify the bias. The sample is biased because it only includes students who attended a basketball game. These students may be more likely to be athletes or fans of basketball, and basketball players tend to be taller than average. Therefore, the sample overrepresents taller students. Step 3: Explain the consequence. Because the sample is not representative of the entire student body (which includes shorter students who did not attend the game), the estimated average height will likely be higher than the true average height of all students. This is called convenience sampling bias. The answer is: Convenience sampling bias; the sample is not representative of the entire student population because it only includes students who attended a basketball game, who are likely taller than average.

4. Liam is a school board member investigating whether the new later start time of 9:00 AM has improved attendance rates among Grade 10 students at a large high school. He obtains a list of all 400 Grade 10 students and sends an online survey to every student, asking them to report whether they have had fewer than five unexcused absences this semester. Only 80 students respond to the survey. Of the respondents, 75 report having fewer than five unexcused absences. Liam concludes that 93.75% of all Grade 10 students have good attendance. Identify and explain two distinct types of bias present in this sampling method, and discuss how each bias could affect the accuracy of Liam's conclusion for the entire Grade 10 population. Answer: Voluntary response bias and non-response bias. Voluntary response bias: students who choose to respond may have better attendance and are more likely to report it. Non-response bias: the 80% who did not respond may have different attendance patterns, likely worse, leading to an overestimate of good attendance. Both biases mean the results cannot be generalized to all 400 students. Solution: Identify the sampling method. Liam sent an online survey to all 400 Grade 10 students, but only 80 chose to respond. This is a voluntary response sample because participation is self-selected.
   Full step-by-step solution

   Step 1: Identify the sampling method. Liam sent an online survey to all 400 Grade 10 students, but only 80 chose to respond. This is a voluntary response sample because participation is self-selected. Step 2: Identify the first type of bias. Voluntary response bias (self-selection bias): students who feel strongly about attendance or who have good attendance are more likely to respond. Those with poor attendance may ignore the survey, skewing the results. Step 3: Identify the second type of bias. Non-response bias: only 20% of the students responded (80 out of 400). The 320 non-respondents (80%) may have systematically different attendance patterns. Students with more absences may be less likely to complete a survey about attendance, leading to underrepresentation of this group. Step 4: Analyze the effect on the conclusion. Liam's reported rate of 93.75% (75/80) is based only on respondents, who likely have better-than-average attendance. The true proportion of all 400 students with fewer than five unexcused absences is almost certainly lower, because the non-respondents likely include more students with poor attendance. Step 5: State the final answer. The two biases are voluntary response bias and non-response bias. Both biases cause the sample to overestimate the proportion of students with good attendance, making Liam's conclusion unreliable for the entire Grade 10 population.

5. Mere wants to estimate the average number of hours per week that students at her school spend on homework. She posts a survey on the school's social media page and receives responses from 84 students. Identify the sampling method used, the type of bias present, and explain why this bias is a problem. Answer: Voluntary response sample; non-response bias; the sample is not representative of all students because it only includes those who choose to respond, likely overrepresenting students with strong opinions or more free time. Solution: Identify the sampling method. Mere posts a survey on social media and waits for students to respond. Step 2: Identify the bias.
   Full step-by-step solution

   Step 1: Identify the sampling method. Mere posts a survey on social media and waits for students to respond. This is a voluntary response sample because participants self-select. Step 2: Identify the bias. The bias is non-response bias (a form of selection bias) because only students who see the post and choose to respond are included. Step 3: Explain why it is a problem. Students who are very busy with homework may not have time to respond, while those with little homework may be more likely to respond. Also, not all students check social media equally. This means the sample is not representative of the entire school population, so the estimated average hours will be inaccurate. The answer: Voluntary response sample; non-response bias; the sample is not representative of all students because it only includes those who choose to respond, likely overrepresenting students with strong opinions or more free time.