Statistical Bias and Sampling

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Answer Key & Explanations

Statistical Bias and Sampling · Grade 10 · Worksheet 3

1. Matiu is a researcher studying the average weekly screen time of high school students in his city. He posts a link to an online survey on his social media page and asks his followers to participate. Within a week, 240 students complete the survey. The average screen time reported by these respondents is 42 hours per week. Identify and explain two distinct types of bias present in Matiu's sampling method, and discuss how each bias could affect the generalizability of his results to all high school students in the city. Answer: Voluntary response bias and selection bias. The sample is not representative because participants self-select (voluntary response bias) and are limited to Matiu's social media followers (selection bias), who likely have higher screen time than the general student population. Solution: Identify the first type of bias — voluntary response bias. Matiu posted a link to an online survey on his social media page and asked his followers to participate. Because of voluntary response bias, the 240 respondents likely have different screen time habits than non-respondents.
   Full step-by-step solution

   Step 1: Identify the first type of bias — voluntary response bias. Matiu posted a link to an online survey on his social media page and asked his followers to participate. This means that students chose whether or not to respond. Voluntary response samples tend to attract people with strong opinions or high interest in the topic, which often leads to overrepresentation of extreme views or behaviors. In this case, students with very high or very low screen time might be more motivated to respond, making the average unreliable. Step 2: Identify the second type of bias — selection bias. The survey is only accessible to Matiu's social media followers, who are not representative of all high school students in the city. For example, Matiu's followers may be from the same school, age group, or socioeconomic background. Additionally, students who do not use social media or do not follow Matiu have no chance to participate. This systematically excludes certain segments of the population, making the sample unrepresentative. Step 3: Explain how these biases affect generalizability. Because of voluntary response bias, the 240 respondents likely have different screen time habits than non-respondents. Because of selection bias, the sample excludes students who are not Matiu's social media followers. Together, these biases mean that the average of 42 hours per week cannot be generalized to all high school students in the city. The true average for the whole population is likely different, but the direction and magnitude of the difference are unknown.

2. Mere is a community health researcher studying physical activity levels among adults in her city. She posts a survey link on a popular fitness forum and receives 420 responses. Of these, 85% report exercising at least 30 minutes daily. Identify and explain the most significant type of sampling bias present in this study, and describe how it affects the generalizability of the results to all adults in the city. Answer: Voluntary response bias (also known as self-selection bias). The sample consists only of people who choose to respond to an online survey posted on a fitness forum, which attracts individuals who are already interested in exercise and likely have higher activity levels than the general adult population. This means the results cannot be generalized to all adults in the city because the sample is not representative. Solution: Identify the sampling method. Mere posted a survey link on a fitness forum, and people chose whether or not to respond. The reported 85% who exercise at least 30 minutes daily is almost certainly an overestimate for the entire adult population of the city.
   Full step-by-step solution

   Step 1: Identify the sampling method. Mere posted a survey link on a fitness forum, and people chose whether or not to respond. This is a voluntary response sample, not a random sample. Step 2: Identify the bias. In voluntary response samples, individuals who feel strongly about the topic are more likely to participate. Here, the forum is specifically about fitness, so its members are already more health-conscious than the average person. Additionally, those who choose to click the link and complete the survey may be particularly proud of their exercise habits. Step 3: Explain how the bias affects generalizability. The reported 85% who exercise at least 30 minutes daily is almost certainly an overestimate for the entire adult population of the city. Adults who do not use fitness forums, who exercise less, or who are not interested in the topic are systematically excluded from the sample. Therefore, the results only reflect the views of a self-selected, fitness-oriented subgroup, not the broader population. Final answer: The most significant bias is voluntary response bias (self-selection bias). The sample is not representative because it only includes fitness forum users who chose to respond, leading to overestimated exercise levels that cannot be generalized to all adults in the city.

3. 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.

4. 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.

5. Aroha is a school board member investigating whether the new school uniform policy has improved student punctuality. She obtains a list of all 1,200 students enrolled at Northwood High School and randomly selects 150 students to survey about their arrival time. However, she distributes the survey only during the morning homeroom period, which starts at 8:30 AM. Of the 150 selected students, 98 complete the survey. Aroha finds that 92% of respondents report arriving before 8:30 AM. Identify and explain two distinct types of bias present in Aroha's sampling method, and discuss how each bias could affect the conclusion that the uniform policy improved punctuality. Answer: The two distinct types of bias are non-response bias (only 98 out of 150 responded) and selection bias due to timing (surveying only during homeroom misses students who are late). Non-response bias could overestimate punctuality because punctual students are more likely to be present. Selection bias could also overestimate punctuality because chronically late students are absent during homeroom and excluded from the sample entirely. Solution: Aroha started with a list of all 1,200 students and randomly selected 150. However, she distributed the survey only during the morning homeroom period (8:30 AM).
   Full step-by-step solution

   Step 1: Understand the sampling process. Aroha started with a list of all 1,200 students and randomly selected 150. However, she distributed the survey only during the morning homeroom period (8:30 AM). Only 98 out of 150 selected students completed the survey, meaning 52 did not respond. Step 2: Identify the first type of bias — non-response bias. Non-response bias occurs when the individuals who do not respond differ systematically from those who do. Here, 52 out of 150 (about 34.7%) did not respond. Students who are frequently late or absent may be less likely to be present during homeroom to complete the survey. If late students are underrepresented among respondents, the reported punctuality rate (92%) would be artificially high, overstating the policy's effectiveness. Step 3: Identify the second type of bias — selection bias due to timing. By distributing the survey only during homeroom at 8:30 AM, Aroha systematically excludes any student who arrives after 8:30 AM. These chronically late students are never given a chance to respond. Since the survey is about punctuality, excluding late students entirely guarantees that the sample will overrepresent punctual students, biasing the results toward a positive conclusion about the uniform policy. Step 4: Summarize the biases and their effects. Both biases work in the same direction: they make the sample appear more punctual than the full student population. Non-response bias excludes late students who are absent, and selection bias excludes late students who arrive after homeroom. The conclusion that the uniform policy improved punctuality would be unreliable because the data come from a biased sample. Final answer: The two distinct types of bias are non-response bias (only 98 out of 150 responded) and selection bias due to timing (surveying only during homeroom misses students who are late). Non-response bias could overestimate punctuality because punctual students are more likely to be present. Selection bias could also overestimate punctuality because chronically late students are absent during homeroom and excluded from the sample entirely.