Statistics Practice Problems

Population, sample, and bias are three of the most commonly misused words in an intro statistics unit, mostly because students learn the definitions in isolation and never have to apply all three to the same scenario. This set of statistics practice problems fixes that by building every question around one running example: a bank CEO who wants to survey his financial advisors and has to decide how to pick who gets asked.

The activity opens with a worked example instead of a bare definition. Students meet Ryan, a bank CEO who wants to survey 50 of his 250 financial advisors, and see how the terms "population" and "sample" map onto his actual survey before they're asked to identify those terms anywhere else. A short reference table follows with plain-language definitions of population, sample, and bias side by side with a worked example of each, so students always have something concrete to point back to instead of memorizing a phrase.

From there, the practice problems shift to identification and diagnosis rather than computation. Students read four short scenarios — a bank manager surveying coworkers about commuting, an employee polling strangers at a park, a coworker generalizing from a lunch order, a manager asking about a party on a specific day — and have to name the sample and population in each one. The last two scenarios are deliberately more subtle than the first two, since spotting sampling bias gets harder once the survey looks reasonable on the surface. The set closes with a two-part problem where students not only identify a flawed sample but have to propose a fix, which is a meaningfully harder ask than just labeling the flaw.

This is a strong fit for the start of a statistics unit, before students touch any actual data collection or calculation. Treating these ideas correctly from day one saves a lot of confusion later, when the same vocabulary comes back attached to real numbers.

This activity is from Module 21 of the Rapunzl curriculum, Financial Statistics.

Introducing Sampling Methods & Bias

In this activity, we explore how gathering data from real-life events can help make predictions about future events. Surveys are usually used to collect statistical data. Since it is not practical or possible to survey an entire population in many situations, a representative sample of the population is selected. There are a variety of sampling techniques, each with its strengths and weaknesses. The choice of sampling technique depends on the population and the type of survey being conducted.

Let’s use a statistical problem to learn about specific terms related to statistics.

Problem: There are 250 employees at Rapunzl Bank who are working as financial advisors. Ryan, the CEO, wishes to survey 50 of these financial advisors. He has chosen 15 senior advisors, 25 junior advisors, and 10 intermediate advisors.

How many financial advisors are beating the expected returns for advisors at Rapunzl Bank? How many hours on average are spent on researching new investments?

1. In each situation, identify the sample and the population.

(a) A bank branch manager surveyed their coworkers at their bank about how they commuted to work to determine what was the most popular form of commuting at all branches of the bank.

(b) John works for an investment firm. He stands at a busy park asking people what types of investment products they use.

(c) Jamal noticed that most of his coworkers were ordering lunch to eat from the famous pizza shop on the corner. He suggested to his boss that the company should do a group order to reduce delivery fees.

(d) Sarah is the manager at a bank. She asks her coworkers on Saturday if they should have an end of quarter party..

1. Sasha wants to know what investment product her coworkers think is best currently. Sasha’s friend, Martha, works on a different floor than her, so they each survey coworkers on their floor. Sasha uses the completed surveys from both floors to draw her conclusions about the best investment product. There are 5 floors at the company.

(a) Identify the population and the sample.

(b) Is the sample representative of the population? If it is representative, explain why. If not, suggest how the sample could be improved.

Teacher Notes

Question 1 has four scenarios, and they're not equally difficult. Parts (a) and (b) are fairly clean cases of population versus sample. Parts (c) and (d) are where students tend to slow down, because both scenarios involve a person drawing a conclusion from an informal, non-random observation rather than a deliberately designed survey. Push students to articulate what the "population" would even be in a case like Jamal noticing his coworkers' lunch orders — there wasn't a formal sample at all, which is itself worth surfacing as a class discussion.

Question 2 is the hardest problem in the set and is worth the most class time. Students need to recognize that surveying only two of five floors, even with real survey data from real employees, still produces a sample that may not represent the whole company. Listen for whether students can articulate why 5 floors specifically matters, not just that "more data would be better" — the goal is for them to name what's structurally missing from Sasha's method, and to propose a concrete fix, such as sampling from all five floors.

This worksheet works well as a warm-up before your class collects any of its own data, since the vocabulary sticks better once students have practiced spotting bias in someone else's flawed process first.

The answer key for this worksheet, along with the rest of the activities in this unit, is in the Rapunzl teacher portal.

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