Study design and inference — Digital SAT Math practice
Sampling, margin of error and what a study can conclude.
Key terms
- Target Population: The entire group about which the researcher wants to draw conclusions from the sample data.
- Random Sample: A selection method where every member of the target population has an equal chance of being chosen.
- Margin of Error: The maximum expected difference between the sample estimate and the true population parameter.
- Confidence Interval: The range formed by adding and subtracting the margin of error from the point estimate.
- Cause-and-Effect: A conclusion that only a randomized experiment can support, showing one variable directly influences another.
- Association: A statistical relationship between variables found in observational studies, which does not prove causation.
Common traps
- Assuming an observational study proves cause and effect just because two variables are strongly correlated.
- Generalizing results from a convenience sample to a whole population without random selection.
- Believing a larger sample fixes bias that comes from a poor sampling method.
Key ideas to know
Confidence Interval Calculation
A survey of 400 voters finds that 62% support Candidate A. If the margin of error is 5%, what is the confidence interval for the true proportion of supporters?
The interval is 57% to 67%. You calculate it by subtracting and adding the margin of error to the sample estimate: 62% ± 5%. Any value between these bounds is a plausible value for the population proportion.
Watch out: Adding the margin of error only in one direction or forgetting that the interval represents plausible values for the entire population, not just the sample.
Cause and Effect Inference
Researchers want to determine if a new fertilizer actually increases crop yield. Which study design should they use, and what conclusion is supported?
They must use a randomized experiment with a treatment group and a control group. Only this design supports a cause-and-effect conclusion about the fertilizer impact on yield.
Watch out: Assuming correlation implies causation just because two variables change together in an observational study or survey.
Limits of Generalization
A researcher takes a simple random sample of 600 registered voters in Ohio to estimate voter preference for an upcoming election. What conclusion is this study design able to support?
The results can be generalized to all registered voters in Ohio because the sample was randomly drawn from that specific target population. Random sampling allows statistical inference about the defined group.
Watch out: Assuming a convenience sample or a sample from a different demographic supports conclusions about the original target population.
Sample Size and Margin of Error
A student claims that doubling the sample size from 1,000 to 2,000 will cut the margin of error in half. Is this claim correct?
No. While a larger random sample does reduce the margin of error, it does not decrease linearly. The margin of error is inversely proportional to the square root of the sample size, so doubling the sample only reduces it by roughly 1.41.
Watch out: Believing that margin of error decreases at the same rate as the sample size increases, rather than following a square root relationship.
Sample questions
Question 1
A researcher selected 250 households at random from a city and found that 62% had at least one pet. Which conclusion is best supported by this result?
- Exactly 62% of all households in the city have at least one pet.
- It is plausible that about 62% of all households in the city have at least one pet.
- 62% of households in every neighborhood of the city have at least one pet.
- Owning a pet causes households to remain in the city.
Show answer and explanation
Answer: It is plausible that about 62% of all households in the city have at least one pet.
A random sample supports a generalization to the population, but only as an estimate — not an exact figure, not neighborhood-by-neighborhood, and never a causal claim.