On the test
Problem-Solving and Data Analysis is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
The way I think about these questions starts with a village rumour. Someone says, 'Every time the rooster sings, the sun comes up. So the rooster makes the sun rise.' Everyone laughs, because we know the rooster and the sun are connected in time, but the rooster does not cause the sunrise. Both happen because of the morning. To be sure that something really causes something else, you would have to test it fairly, like taking away the rooster and seeing if the sun still comes. On the SAT, I ask two questions. Who was picked for the study, so who can the result be about? And did the researchers decide who got the treatment by chance, so can we say it caused the change?
These questions use only words, and the rules are few. Once you know the two checks, who was picked and how groups were formed, you can answer almost all of them.
What is inside this part
Studies where researchers only collect information and do not give a treatment, so they can show a link but not a cause.
Surveys and records · Association versus causation · Confounding factors
Studies where researchers give a treatment to some people and compare with others, ideally using random assignment.
Treatment and control groups · Placebo · Random assignment
Deciding to which group the results can be applied, based on how the people were chosen.
Random sample versus volunteers · Same population only · Nonresponse
Picking the statement that matches the study design without claiming too much or too little.
Words like causes and is associated with · Best study design for a claim
How to know a question is from this part
A description of how researchers chose people and what they did, followed by 'Which conclusion is best supported?' or 'Which study design...?'.
- 'Randomly selected', 'volunteers', 'chose to', or 'responded'
- 'Randomly assigned' to groups, or a placebo or control group
- 'Which conclusion is best supported / justified'
- 'Which of the following would allow researchers to conclude...'
- 'Observational study', 'experiment', or 'is associated with'
The lesson
1. Two kinds of studies
An observational study just watches or asks. The researchers do not give anything to anyone. They record what already happens, for example by a survey or by looking at past records.
An experiment is different. The researchers give a treatment to some people (like a medicine, a fertilizer, a teaching method) and compare with a group that does not get it. The researchers decide who gets what.
Quick test: did the researchers do something to the people? If yes, experiment. If they only collected information, observational.
2. Random sampling: who can the result be about?
A random sample means the people in the study were chosen by chance from a larger population. Every member had an equal chance. Because of this, the sample is likely to look like the population.
So the first check is: was the sample random? If yes, the result can be applied to the whole population it came from. If the people were volunteers, or only came from one place or group, the result applies only to people like them.
Keep the population exact. A random sample from one school applies to that school, not to all schools.
300 students are randomly selected from a school of 2,400. Results apply to which group?
Show the steps
- The sample is random.
- It came from the school of 2,400.
- So the results apply to all 2,400 students, not to other schools.
All 2,400 students
3. Random assignment: can we say it caused the change?
Random assignment means the people in the study are put into groups (treatment or no treatment) by chance, like a lottery. It only happens in experiments.
Why? Chance spreads other differences, such as age, health and motivation, evenly among the groups. Then if the groups end up different, the treatment is the most likely reason. This is how we show a cause.
The group that gets no real treatment is the control group. Sometimes it gets a placebo, a fake treatment like a sugar pill, so that nobody changes because of just believing.
60 volunteers are randomly assigned to use a new fertilizer or not. What is random assignment for?
Show the steps
- It makes the two groups similar.
- So a difference in plant growth can be linked to the fertilizer.
- It is not about applying to all farmers.
To link the difference to the treatment
4. The two-check table
I always put the study through two checks. Check 1: random sample? Check 2: random assignment?
Random sample = yes: the result can be applied to the whole population. No: only to the people in the study.
Random assignment = yes: a cause can be claimed. No: only an association (a link) can be claimed.
If both are yes, the strongest conclusion is allowed: the treatment caused the change in the population. If both are no, the weakest: a link among those studied only.
80 adult volunteers from a city are randomly assigned to a medicine or a sugar pill. The medicine group improves faster, beyond chance. What conclusion?
Show the steps
- Random sample? No, volunteers.
- Random assignment? Yes.
- Cause is supported, but only for people like these volunteers, not all adults in the city.
Medicine probably caused it, for volunteers like these
5. Correlation is not causation
Two things can go up together without one causing the other. Ice cream sales and drownings both go up in hot months. The cause of both is hot weather. Hidden causes like this are called confounding factors.
In an observational study, you can never rule out a confounding factor. So the correct conclusion is 'is associated with' or 'is linked to', not 'causes'.
Watch for the words 'self-selected' or 'chose to join'. When people choose their own group, motivated or healthier people might choose the treatment, and that, not the treatment, may explain the difference.
Students who chose a tutoring program gained more than those who did not. Does tutoring cause the gain?
Show the steps
- Students chose their group, so no random assignment.
- Motivated students might both join and gain more.
- So only an association can be claimed.
No, only an association
6. Other problems with a study
Nonresponse: if you pick a random sample but only some answer, those who answer may be different. Then the result may not match the population even though the sample was random at the start.
Bias in the question or the sampling: for example surveying only people at a gym about exercise.
Small samples: a very small sample gives a large margin of error, but the SAT mostly tests the design, not size.
Also, 'statistically significant' or 'unlikely to be due to chance' means the difference is probably real, but you still need the two checks for who it applies to and whether it caused the change.
7. Choosing the best conclusion
Read the options and cross out the ones which claim too much. 'Causes' needs random assignment. 'All students' or 'all adults' needs a random sample from that group.
Also cross out the ones which claim too little. If the study had both random sampling and random assignment, 'cannot show a cause' is wrong.
The right answer is usually the one that matches the study exactly: the right population and the right word, causes or is associated with.
200 gym members are randomly selected, then randomly assigned to a new or usual plan. The new-plan group lost more weight, beyond chance. Conclusion?
Show the steps
- Random sample of gym members: yes, so applies to the gym members.
- Random assignment: yes, so a cause is supported.
- The new plan caused more weight loss for members of this gym.
Caused, for members of this gym
See it
| Random assignment: YES | Random assignment: NO | |
|---|---|---|
| Random sample: YES | The treatment caused the change, for the whole population | A link (association) for the whole population, not a cause |
| Random sample: NO | The treatment caused the change, for people like those studied | A link for only the people studied, not a cause |
- 1Find the population the question is about.
- 2Check 1: were the people chosen at random from it?
- 3Check 2: were people put into groups by chance?
- 4Match the words: 'caused' needs check 2; 'all of them' needs check 1.
- 5Pick the option that matches the study, no more and no less.
- How the people are picked from the population.
- Gives the right to apply results to the population.
- Happens in surveys and experiments.
- How the people are put into groups.
- Gives the right to claim a cause.
- Happens only in experiments.
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