Study Design and Sampling: ACT Math Study Guide

How data is collected determines what conclusions can honestly be drawn from it. In a simple random sample, every member of the population has an equal chance of selection. More generally, probability-based random sampling uses a chance process instead of convenience or self-selection and can support generalizing to the population represented by the sampling frame. A convenience sample draws from whoever is easiest to reach and risks bias. A voluntary response sample only includes those who choose to respond, which tends to overrepresent people with strong opinions.

Only a randomized experiment can support a causal conclusion. An observational study compares existing groups without assigning who goes where, which can reveal an association between two variables but never establishes that one causes the other. Random assignment, not random sampling, is what makes a causal claim possible.

A two-way table organizes data by two categorical variables at once. Each cell where a row and column intersect is a joint frequency, the count of subjects matching both categories. Each row or column total is a marginal frequency, the count for that category alone, regardless of the other variable.

Where this appears as a model: a question about whether a treatment causes an effect. Randomly assigning half of a set of fields to receive a new fertilizer and half to receive none is a randomized experiment, supporting a causal conclusion; simply observing which existing farms already use the fertilizer is only an observational study.

▸ Bias in sampling methods

Worked Example A radio station asks listeners to call in and vote for their favorite song. What kind of sample is this, and what bias does it risk?

Only listeners who choose to call in are included. This is a voluntary response sample.

It risks overrepresenting listeners with strong opinions, since people with a mild preference are less likely to call in at all.

▸ Reading a two-way table

A cell where a row and column intersect gives a joint frequency; the count for a category alone, found in a row or column total, is a marginal frequency. A conditional proportion restricts the calculation to just one row or one column instead of the grand total.

▸ Practice

Question 1

A city wants to know residents' opinions and surveys everyone who visits the city website in one week. What kind of sample is this?

A. Random sample
B. Stratified sample
C. Census
D. Convenience sample

Question 2

A study compares test scores between students who already chose to attend tutoring and students who did not, without random assignment. What kind of study is this, and what can it conclude?

F. Randomized experiment; causation
G. Observational study; causation
H. Observational study; association only
J. Randomized experiment; association only

Question 3

In a two-way table, what does a cell where a row and column intersect represent?

A. A marginal frequency
B. A joint frequency
C. A percentage of the grand total only
D. The row total

Questions 4–6 refer to the table below.

	Approve	Disapprove	Total
Under 30	40	10	50
Over 30	25	25	50
Total	65	35	100

Question 4

What proportion of respondents over 30 disapprove?

F. 0.35
G. 0.50
H. 0.25
J. 0.10

Question 5

What proportion of all respondents approve?

A. 0.65
B. 0.40
C. 0.50
D. 0.25

Question 6

A researcher concludes that being under 30 causes higher approval, based on this table alone. Is that conclusion supported?

F. Yes, the data proves it
G. No, this is an observational comparison, not a randomized experiment
H. Yes, because the sample size is large enough
J. No, because the table does not add up correctly

Question 7

A teacher surveys only the students who volunteer to stay after class. What kind of sample is this?

A. Voluntary response sample
B. Random sample
C. Systematic sample
D. Stratified sample

Question 8

Researchers compare outcomes between patients who chose surgery and those who chose medication, without random assignment. What can this study conclude?

F. Observational study; causation
G. Randomized experiment; causation
H. Randomized experiment; association only
J. Observational study; association only

Question 9

A survey asks, “Don't you agree that the new park is a great addition?” What is the main issue with this question?

A. No issue, it's a fair question
B. The sample size is too small
C. Leading question bias
D. Random sampling error

Questions 10, 11, and 14 refer to the table below.

	Yes	No	Total
Group A	28	22	50
Group B	34	16	50
Total	62	38	100

Question 10

How many people in Group A answered “Yes”?

F. 22
G. 34
H. 16
J. 28

Question 11

Find the overall proportion of all respondents who answered “Yes.”

A. 0.56
B. 0.62
C. 0.44
D. 0.38

Question 12

A city surveys residents by calling only landline phone numbers. What is the main risk?

F. No risk at all
G. Overrepresenting younger residents
H. Underrepresenting residents without landlines
J. A guaranteed random sample

Question 13

What does random assignment, as opposed to random sampling, primarily support?

A. A causal conclusion between groups
B. Generalizing results to a larger population
C. Reducing the needed sample size
D. Eliminating all bias entirely

Question 14

Find the probability that a randomly chosen respondent from Group A answered “Yes.”

F. 0.62
G. 0.56
H. 0.44
J. 0.38

Question 15

A pollster surveys shoppers exiting one specific store on one afternoon. What kind of sample is this?

A. Random sample
B. Stratified sample
C. Systematic sample
D. Convenience sample

Question 16

Researchers observe existing exercise habits and compare cholesterol levels across groups, without assigning who exercises. What can this study conclude?

F. Observational study; association only
G. Observational study; causation
H. Randomized experiment; causation
J. Randomized experiment; association only

Questions 17, 18, and 20 refer to the table below.

	Yes	No	Total
Group A	32	18	50
Group B	21	29	50
Total	53	47	100

Question 17

How many people in Group A answered “Yes”?

A. 18
B. 32
C. 21
D. 29

Question 18

Find the overall proportion of all respondents who answered “Yes.”

F. 0.53
G. 0.50
H. 0.47
J. 0.58

Question 19

A survey is conducted by mailing paper forms only to registered voters with a listed home address. What is the main risk?

A. No risk at all
B. Overrepresenting renters
C. Underrepresenting voters without a listed address
D. A guaranteed random sample

Question 20

Find the probability that a randomly chosen member of Group B answered “Yes.”

F. 0.42
G. 0.53
H. 0.58
J. 0.29

▸ Answers and Explanations

Show the answers and explanations

1: D. Only residents who chose to visit the website in that week could respond, making this a convenience sample instead of one giving every resident an equal chance. Choice A assumes every resident had an equal chance of being surveyed, without recognizing that only self-selected website visitors could respond.

2: H. Without random assignment, any difference between the groups could be explained by other factors, so only an association, not causation, can be concluded. Choice F mistakes this for a randomized experiment, overlooking that students chose their own group instead of being randomly assigned.

3: B. A cell where a row and column intersect counts subjects matching both categories at once, a joint frequency. Choice A confuses a joint frequency, found at the intersection of a row and column, with a marginal frequency, found in a row or column total.

4: G. 25 out of the 50 over-30 respondents disapprove: 25/50 = 0.50. Choice F uses the wrong total, matching the disapprove count against the grand total instead of just the over-30 total.

5: A. 65 out of 100 respondents approve overall: 65/100 = 0.65. Choice B reports only the under-30 approve count over the grand total, an incomplete calculation.

6: G. This is an observational comparison of two existing groups, not a randomized experiment, so causation cannot be established from this data alone. Choice F mistakes this observational comparison for a randomized experiment, which is required to support a causal claim.

7: A. Only students who choose to stay respond, making this a voluntary response sample. Choice B assumes every student had an equal chance of being surveyed, without recognizing that only volunteers actually respond.

8: J. Without random assignment, any difference between the groups could be explained by other factors, so only an association can be concluded. Choice F correctly identifies this as observational but incorrectly claims causation can be concluded without random assignment.

9: C. The question's phrasing pushes respondents toward a specific answer, a leading question. Choice A fails to recognize that the question's phrasing pushes respondents toward a specific answer.

10: J. Directly given: 28 people in Group A answered Yes. Choice G reports Group B's count instead of the requested Group A count.

11: B. Total Yes: 28+34=62; grand total: 100; 62/100=0.62. Choice A reports Group A's individual proportion instead of the requested overall proportion.

12: H. Residents without a landline (often younger or lower-income households) would have no chance of being reached, risking underrepresentation. Choice F fails to recognize that residents without a landline have no chance of being reached by this method.

13: A. Random assignment to groups is what allows a difference in outcomes to be attributed to the treatment itself, supporting a causal conclusion. Choice B confuses the purpose of random assignment with the purpose of random sampling, which supports generalizing results instead.

14: G. Restricting to Group A: 28 Yes out of 50 total, 28/50=0.56. Choice F reports the overall proportion instead of the proportion restricted to Group A.

15: D. Only shoppers present at that store during that specific window could be surveyed, making this a convenience sample. Choice A assumes every shopper had an equal chance of being surveyed, without recognizing that only shoppers present at that specific time and place could respond.

16: F. Without random assignment, any difference in cholesterol levels could be explained by other factors, so only an association, not causation, can be concluded. Choice G correctly identifies this as observational but incorrectly claims causation can be concluded without random assignment.

17: B. Directly given: 32 people in Group A answered Yes. Choice A reports the “No” count for Group A instead of the requested “Yes” count.

18: F. Total Yes: 32+21=53; grand total: 100; 53/100=0.53. Choice G comes from an arithmetic slip while combining the two groups' “Yes” counts.

19: C. Voters without a listed home address, or without a permanent address at all, would have no chance of being reached, risking underrepresentation. Choice A fails to recognize that voters without a listed address have no chance of being reached by this method.

20: F. Restricting to Group B: 21 Yes out of 50 total, 21/50=0.42. Choice G reports the overall proportion instead of the proportion restricted to Group B.

Study Design and Sampling

▸ Bias in sampling methods

▸ Reading a two-way table

▸ Practice

▸ Answers and Explanations

PART V · STATISTICS & PROBABILITY