Question 45·Hard·Evaluate Statistical Claims: Observational Studies and Experiments
To estimate the mean daily time that teenagers in a city spend on social media, a marketing firm used a computer to randomly dial landline telephone numbers. The firm successfully surveyed 600 households that had at least one teenager. For the teenagers surveyed, the mean reported daily social-media use was 2.9 hours.
Which statement is most appropriate based on this information?
For questions about surveys and experiments, first identify the population (who we care about) and the sample (who was actually studied). Then, look closely at how the sample was chosen: ask whether any groups were systematically left out or had different chances of being included. Remember that a large sample does not guarantee an unbiased estimate and that random selection must apply to the whole population, not just a subset like landline owners. Finally, eliminate answer choices that claim certainty about the population mean or that overstate what the sampling method can justify.
Hints
Think about who was included in the sample
Focus on the phrase "randomly dial landline telephone numbers." Which teenagers in the city could not be reached by this method?
Population vs. sample
Ask yourself: Does a sample mean always equal the population mean, even with a good sampling method? What is usually the correct way to interpret a sample mean?
Does a large sample fix everything?
Consider whether simply having 600 responses guarantees that the sample is representative. What if certain groups of teenagers never had a chance to be in the sample at all?
Random but from what group?
Random dialing can help, but only within the group you are dialing. Think about whether dialing random landline numbers is the same as giving every teenager in the city an equal chance to be selected.
Step-by-step Explanation
Identify the goal and the data collected
The goal is to estimate the mean daily time on social media for all teenagers in the city (the population mean).
What we actually have is a sample: 600 households that both (1) have a landline and (2) have at least one teenager. From those teenagers, the sample mean daily use is 2.9 hours.
Understand what makes a sample unbiased
An estimate is unbiased when the sampling method gives every individual in the population a similar chance of being selected, and there is no systematic reason the sample would differ from the population.
If some types of teenagers have no chance or a much smaller chance of being selected, the sample can be biased, and the sample mean might systematically differ from the true population mean.
Analyze the sampling method used
The firm randomly dialed landline numbers and then surveyed households that had at least one teenager.
Think about who is left out:
- Households that do not have landline phones (for example, cell-phone-only households) are never called.
- Teenagers in those households have zero chance of being included.
If teenagers in households without landlines use social media differently (for example, possibly more if they are in more tech-focused or lower-cost cell-only homes), then the sample mean of 2.9 hours could be systematically different from the true city mean.
Evaluate each answer choice
Now match this reasoning to the choices:
- Choice A claims the population mean is 2.9 hours. A sample statistic is only an estimate; with possible bias in sampling, we cannot assert the true mean equals 2.9.
- Choice B says a large sample size alone makes the estimate unbiased. This is false: a large biased sample is still biased (it just estimates the wrong value more precisely).
- Choice D says the method is unbiased because landline telephone numbers were dialed at random. But the randomness only applies within the group of landline households; it does not include households without landlines.
State the conclusion
Because teenagers in households without landline phones were completely excluded and might have different social-media habits, the sample may not represent all teenagers in the city. Therefore, the most appropriate statement is: The sampling method may be biased because households without landline telephones were not included.