Statistics: Sampling
Designing and evaluating samples for statistical studies
Statistics: Sampling
Designing and evaluating samples for statistical studies
Statistics - Grade 9-12
- 1
A school wants to estimate the average number of hours students spend on homework each week. The principal surveys 50 students from the advanced placement study hall. Identify the population, the sample, and one possible source of bias.
The population is the full group the school wants information about.
The population is all students at the school. The sample is the 50 students from the advanced placement study hall. A possible source of bias is that students in that study hall may have different homework habits than the average student at the school. - 2
A city divides its residents into five age groups and randomly selects 100 people from each age group to survey about public transportation. What sampling method is being used, and why is it useful?
This is stratified random sampling because the population is divided into age groups, and a random sample is selected from each group. It is useful because it helps make sure each age group is represented in the survey. - 3
A researcher assigns every student in a school a number and uses a random number generator to choose 120 students for a survey. What sampling method is this?
Look for whether every member of the population has an equal chance of selection.
This is a simple random sample because every student has an equal chance of being selected by the random number generator. - 4
A mall employee asks the first 80 shoppers who walk past a booth to answer a question about a new store. What sampling method is being used, and what is one concern about it?
This is a convenience sample because the employee is surveying people who are easy to reach. One concern is that shoppers who pass that booth may not represent all mall shoppers. - 5
A teacher wants to know whether students prefer online or paper textbooks. She writes every student's name on a slip of paper, mixes the slips, and draws 30 names. Explain why this method could produce an unbiased sample.
Random selection helps avoid personal choice affecting the sample.
This method could produce an unbiased sample because each student has an equal chance of being chosen. Random selection helps reduce the chance that the teacher chooses only certain types of students. - 6
A survey about student cafeteria food is posted online, and any student who wants to respond may do so. What type of sample is this, and why might the results be biased?
This is a voluntary response sample. The results might be biased because students with strong opinions, especially very positive or very negative opinions, may be more likely to respond. - 7
A town has 60 percent adults and 40 percent children. A survey about park improvements includes 90 adults and 10 children. Explain why this sample may not represent the town well.
Compare the population percentages to the sample percentages.
The sample may not represent the town well because adults are overrepresented and children are underrepresented. A more representative sample would have proportions closer to 60 percent adults and 40 percent children. - 8
A factory produces light bulbs on 12 machines. An inspector randomly selects 5 bulbs from each machine every hour to test. What sampling method is used, and why might it be better than sampling from only one machine?
This is stratified sampling because bulbs are sampled from each machine. It is better than sampling from only one machine because it can detect problems that affect specific machines and gives a more complete picture of production quality. - 9
A researcher wants to survey households in a large city. She randomly selects 20 city blocks and surveys every household on those blocks. What sampling method is this?
In cluster sampling, entire groups are chosen rather than individuals from every group.
This is cluster sampling because the researcher randomly selects groups, or blocks, and then surveys every household within the selected groups. - 10
A school newspaper asks students, 'Do you agree that the new schedule is unfair and should be changed immediately?' Identify the problem with the wording and rewrite the question in a less biased way.
A neutral survey question should not suggest the answer.
The question is biased because it uses words like unfair and should be changed immediately, which may pressure students toward one answer. A less biased question is: Do you support or oppose the new school schedule? - 11
A polling company wants to estimate support for a candidate in a state. It randomly selects 1,000 registered voters from a current voter list and contacts them by phone. Name one strength and one limitation of this sampling plan.
One strength is that the sample is randomly selected from registered voters, which helps reduce selection bias. One limitation is that people who do not answer the phone or refuse to participate may differ from those who respond. - 12
A sample of 25 students is selected from a high school with 2,000 students to estimate the percentage of students who ride the bus. Is the sample size alone enough to decide whether the sample is representative? Explain.
Think about both how many people are sampled and how they are chosen.
The sample size alone is not enough to decide whether the sample is representative. The method of selection also matters because a small random sample can be better than a larger biased sample. - 13
A company has employees in three departments: sales, technology, and customer support. It wants a stratified sample of 60 employees in proportion to department size. Sales has 100 employees, technology has 150 employees, and customer support has 50 employees. How many employees should be selected from each department?
Find each department's fraction of the total, then multiply by 60.
There are 300 employees total. Sales is 100 out of 300, so 20 employees should be selected. Technology is 150 out of 300, so 30 employees should be selected. Customer support is 50 out of 300, so 10 employees should be selected. - 14
A sports website surveys only its subscribers to estimate how many people in the country watch professional basketball. Explain why this sampling frame may cause bias.
The sampling frame may cause bias because subscribers to a sports website are more likely to be interested in sports than people in the general country population. They may watch professional basketball at a higher rate than average. - 15
A researcher surveys 500 randomly selected adults and finds that 58 percent support a new recycling program. The researcher states, 'Exactly 58 percent of all adults support the program.' Explain why this conclusion is too strong.
Sampling always includes some uncertainty.
The conclusion is too strong because a sample statistic is an estimate, not an exact value for the entire population. Random samples can vary, so the researcher should say that about 58 percent of adults in the population may support the program.