Statistical Bias and Sampling

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Answer Key & Explanations

Statistical Bias and Sampling · Grade 10 · Worksheet 2

1. Liam wants to estimate the average height of students at his school. He surveys 50 students from the basketball team. Identify the type of sampling bias present and explain why this sample is not representative of the entire student population. Answer: Selection bias (specifically, undercoverage bias). The sample only includes basketball players, who tend to be taller than the average student, so the sample overrepresents tall students and underrepresents shorter students. Solution: Identify the sampling method. Liam uses a convenience sample by only surveying members of the basketball team. Step 2: Recognize the bias.
   Full step-by-step solution

   Step 1: Identify the sampling method. Liam uses a convenience sample by only surveying members of the basketball team. Step 2: Recognize the bias. This is selection bias, specifically undercoverage bias, because the sample excludes all students who are not on the basketball team. Step 3: Explain why it is not representative. Basketball players are generally taller than the average student due to the nature of the sport. Therefore, the sample will overestimate the average height of all students at the school. Step 4: Conclusion. The sample is biased and cannot be used to accurately estimate the population mean height. The correct answer is selection bias (undercoverage bias).

2. Mere wants to estimate the average height of all students at her school. She posts a survey on the school's social media page and asks students to voluntarily submit their height. Of the 1200 students at the school, only 84 respond. Identify the sampling method used, the type of bias present, and explain why this bias is problematic. Answer: Voluntary response sample; nonresponse bias; the sample is not representative because only those with strong opinions or easy access to social media respond, leading to overrepresentation of certain groups. Solution: Identify the sampling method. Mere posts a survey on social media and asks for volunteers. Since only 84 out of 1200 students responded, the sample suffers from nonresponse bias.
   Full step-by-step solution

   Step 1: Identify the sampling method. Mere posts a survey on social media and asks for volunteers. This is a voluntary response sample, where individuals choose to participate. Step 2: Identify the type of bias. Since only 84 out of 1200 students responded, the sample suffers from nonresponse bias. Those who respond may have different characteristics (e.g., more free time, stronger opinions, or easier internet access) than those who do not respond. Step 3: Explain why this is problematic. The sample is not representative of the entire student body. For example, students without social media access or those less engaged might be excluded, leading to an inaccurate estimate of the average height. The results cannot be generalized to the whole school. Final answer: Voluntary response sample; nonresponse bias; the sample is not representative because only those with strong opinions or easy access to social media respond, leading to overrepresentation of certain groups.

3. A city council wants to estimate the average daily screen time for teenagers in their city. The city has 15,000 teenagers aged 13-18. The council surveys the first 80 students who enter the public library on Saturday morning. If the true population mean is 6.2 hours per day with a standard deviation of 1.8 hours, what is the approximate standard error of the mean for this sampling method? Answer: 0.201 Solution: Identify the formula for standard error of the mean: standard error = population standard deviation / sqrt(sample size) Extract the given values: population standard deviation = 1.8 hours, sample size = 80 students Calculate the square root of the sample size: sqrt(80) = 8.944 Divide the…
   Full step-by-step solution

   Step 1: Identify the formula for standard error of the mean: standard error = population standard deviation / sqrt(sample size) Step 2: Extract the given values: population standard deviation = 1.8 hours, sample size = 80 students Step 3: Calculate the square root of the sample size: sqrt(80) = 8.944 Step 4: Divide the standard deviation by the square root of sample size: 1.8 / 8.944 = 0.201 Step 5: The standard error is approximately 0.201 hours The answer is 0.201.

4. Hana is a school board member investigating whether the new after-school sports program has improved student fitness levels at a high school with 1,200 students. She obtains a list of all students enrolled in the school's physical education classes and sends an email survey to every student on that list, asking them to report their weekly exercise hours and whether they participate in the new program. Only 180 students respond to the survey. Among the respondents, Hana notices that 144 of them participate in the new sports program, and their average weekly exercise hours is 8.4 hours. The remaining 36 respondents do not participate in the new program, and their average weekly exercise hours is 4.2 hours. Identify and explain two distinct types of bias present in this sampling method that could affect Hana's ability to draw valid conclusions about the entire student population's fitness levels and the effectiveness of the new sports program. Answer: Selection bias (sampling only from physical education class lists) and non-response bias (low response rate of 180 out of 1,200 students, with likely overrepresentation of program participants). Solution: Identify the sampling frame. Hana only surveyed students enrolled in physical education classes. Out of 1,200 students on the list, only 180 responded (15% response rate).
   Full step-by-step solution

   Step 1: Identify the sampling frame. Hana only surveyed students enrolled in physical education classes. This excludes students who are not taking PE (e.g., those with medical exemptions, upperclassmen who have completed their PE requirement, or students who opt out). This is selection bias because the sample frame does not represent the entire student population. Step 2: Identify non-response bias. Out of 1,200 students on the list, only 180 responded (15% response rate). Among respondents, 144 out of 180 (80%) participate in the new sports program, while only 36 (20%) do not. It is highly likely that program participants are more motivated to respond because they have a vested interest in the program's success. Non-respondents may include many students who do not participate in the program and have lower exercise hours, leading to an overestimate of overall fitness levels. Step 3: Calculate the average exercise hours among respondents only (if needed for context): Average = (144 * 8.4 + 36 * 4.2) / 180 = (1209.6 + 151.2) / 180 = 1360.8 / 180 = 7.56 hours per week. This average is inflated by the high proportion of program participants. Step 4: Explain how biases affect conclusions. Because the sample is not representative (selection bias) and those who responded are likely more active (non-response bias), Hana cannot generalize the results to all 1,200 students. The true average exercise hours for the entire school is likely lower than 7.56 hours, and the effectiveness of the new program cannot be properly evaluated without a random sample. Final answer: Selection bias (sampling only from physical education class lists) and non-response bias (low response rate of 180 out of 1,200 students, with likely overrepresentation of program participants).

5. Emma is a school board researcher investigating the average weekly study time for Grade 10 students in a large school district. She obtains a list of all 3,500 enrolled Grade 10 students and randomly selects 175 of them to participate in a survey. However, only 63 students complete and return the survey. All 63 respondents are from the district's three specialized STEM magnet schools, which have selective admissions. Identify and explain two distinct types of bias present in this sampling method that could affect the generalizability of the results to the entire Grade 10 population of the district. Answer: The two distinct types of bias are non-response bias (only 63 out of 175 responded, and the non-respondents may have different study habits) and selection bias (all respondents come from selective STEM magnet schools, so they do not represent the broader Grade 10 population, which includes students from regular schools). Solution: Identify the sampling process. Emma started with a list of all 3,500 Grade 10 students in the district and randomly selected 175. This initial random selection should be representative of the whole population.
   Full step-by-step solution

   Step 1: Identify the sampling process. Emma started with a list of all 3,500 Grade 10 students in the district and randomly selected 175. This initial random selection should be representative of the whole population. Step 2: Identify the response rate. Only 63 out of 175 students responded. That means 112 students did not respond (175 - 63 = 112). The response rate is 63/175 = 36%, so nearly two-thirds of the selected students did not participate. Step 3: Identify the first type of bias — non-response bias. Non-response bias occurs when the students who did not respond differ in a meaningful way from those who did respond. For example, students with heavy study loads might be too busy to complete a survey, while those with lighter loads might have more time. Since 112 out of 175 did not respond, the results from the 63 respondents may not reflect the true average study time of all 175 selected students, let alone the entire district. Step 4: Identify the second type of bias — selection bias (specifically, sampling bias within respondents). The problem states that all 63 respondents are from the district's three specialized STEM magnet schools, which have selective admissions. These students likely have different study habits, schedules, and academic pressures compared to students at regular high schools. Since the target population is all Grade 10 students in the district, and STEM magnet students are a small, non-representative subset, the sample no longer represents the broader population. Step 5: State the two distinct biases clearly. 1. Non-response bias — because only 63 out of 175 responded, and the missing 112 may differ systematically in their study time. 2. Selection bias — because all respondents came from selective STEM magnet schools, so the sample does not represent the general Grade 10 population of the district. The answer is: The two distinct types of bias are non-response bias (only 63 out of 175 responded, and the non-respondents may have different study habits) and selection bias (all respondents come from selective STEM magnet schools, so they do not represent the broader Grade 10 population, which includes students from regular schools).