Question 29·Medium·Evaluate Statistical Claims: Observational Studies and Experiments
A university research office randomly selected 150 full-time undergraduate students from the university’s more than 20,000 full-time undergraduates and asked each student, “About how many hours did you spend studying last week?” The mean number of hours reported by the sampled students was 13.4 hours.
Based on these data, which conclusion is most appropriate?
For questions about statistical conclusions, first identify the sample (who was actually studied) and the population we can legitimately generalize to—never go beyond that group. Next, decide whether the statistic given (like a sample mean) should be treated as an exact value or as an estimate that is likely close to the true value; on the SAT, a sample almost always leads to an approximate conclusion with possible error, not an exact one. Use these two checks—correct population and appropriate strength of claim—to quickly eliminate answer choices that overgeneralize or make overly precise statements.
Hints
Focus on who was sampled
Look carefully at the description of the 150 students. Exactly which group of students do they represent: all full-time students at this university, all students at the university, or students nationwide?
Sample mean vs. true mean
The 13.4 hours is an average from the 150 students. Think about whether a sample average is usually exactly the same as the true average for the whole population, or only an estimate of it.
Strength of the conclusion
Ask yourself: based on a random sample, can we make a statement that uses the word “exactly,” or should we allow for some difference between the sample result and the true population value?
Eliminate overly broad claims
Check each answer for whether it talks about a group that goes beyond the group that was actually sampled (for example, including part-time students or students from other universities). Those conclusions are not supported by the data.
Step-by-step Explanation
Identify the sample and the population
First, separate the sample from the population.
- Sample: the 150 full-time undergraduate students at the university who were randomly selected.
- Population we care about: all full-time undergraduate students at that same university (more than 20,000 students).
Any conclusion supported by the data must be about this population, not about other groups.
Understand what the sample mean tells us
The mean for the sample was hours. This is a sample statistic.
- A sample statistic is used to estimate a population parameter (the true average for all students in the population).
- Because the sample is random and reasonably large, the sample mean should be close to the true population mean.
- However, due to sampling variability, we do not expect the sample mean to be exactly equal to the true mean.
Decide whether we can say “exactly” or only “close to”
Now compare two ideas:
- Saying the population mean is exactly hours would ignore sampling variability and margin of error.
- Saying the population mean is likely near hours, but might be a bit higher or lower, correctly reflects that a sample gives an estimate with some margin of error.
So any answer that claims the true average is exactly hours is too strong.
Check which group we are allowed to generalize to and choose the conclusion
Because the sample consists of full-time undergraduates at this university, we can only generalize to all full-time undergraduates at this same university, not to:
- part-time students
- students at other universities
- all full-time undergraduates in the country
The only appropriate conclusion is that the average number of hours spent studying last week by all full-time undergraduate students at the university is likely close to 13.4 hours, but the exact value may differ by some margin of error.