Inference from Sample Statistics and Margin of Error Flashcards

A researcher conducts two independent surveys. Survey A samples 400 people and finds 52% support a policy, with a margin of error of ±4.9%. Survey B samples 900 people and finds 48% support the same policy, with a margin of error of ±3.3%. Which of the following conclusions is statistically justified?

Answer: The surveys are consistent with each other because their confidence intervals overlap.

Survey A's interval is approximately 47.1%–56.9% and Survey B's is approximately 44.7%–51.3%. Because these intervals overlap substantially, both surveys are statistically consistent with the same underlying population proportion. Overlapping confidence intervals mean the difference between estimates is not statistically significant. Choice A confuses a point-estimate difference with a contradiction. Choice D is tempting but wrong: a larger sample makes an estimate more precise, not necessarily closer to the truth — both estimates could be accurate.

A polling organization wants to estimate the proportion of voters who support a ballot measure within ±2 percentage points at a 95% confidence level. They initially plan to survey 2,400 people. If they instead want to cut the margin of error in half (to ±1 percentage point), how many people must they survey?

Answer: 9,600

The margin of error is proportional to 1/√n. To halve the margin of error, you must multiply the sample size by 4 (since 1/√(4n) = 1/(2√n)). Starting from 2,400, the new required sample size is 2,400 × 4 = 9,600. This is a key non-intuitive result: halving precision error requires quadrupling the sample, not doubling it.

A 95% confidence interval for the mean daily screen time of teenagers is reported as (5.8 hours, 7.2 hours). A skeptic claims, 'There is a 5% chance that the true mean is outside this interval.' Which of the following best identifies the flaw in the skeptic's reasoning?

Answer: The true mean is a fixed (non-random) value, so it either is or is not in this specific interval — probability does not apply to it.

This is a classic frequentist vs. Bayesian interpretation trap. In frequentist statistics, the true population mean is a fixed constant — it doesn't move. The interval (5.8, 7.2) either contains the true mean or it doesn't; there's no probability about it. The 95% refers to the procedure: if this method were repeated many times, 95% of the resulting intervals would capture the true mean. The skeptic incorrectly treats the true mean as if it were randomly distributed around the interval.

In a study, researchers randomly sampled 150 students from a high school of 3,000 and found the average GPA to be 3.1 with a standard deviation of 0.6. A second researcher argues that the margin of error should be adjusted using a finite population correction because the sample is a large fraction of the population. The finite population correction factor is √((N−n)/(N−1)). What is the corrected margin of error at 95% confidence (use z* ≈ 2)?

Answer: Approximately ±0.091

The uncorrected margin of error = z* × (s/√n) = 2 × (0.6/√150) = 2 × (0.6/12.25) ≈ 2 × 0.049 ≈ 0.098. The finite population correction factor = √((3000−150)/(3000−1)) = √(2850/2999) = √(0.9503) ≈ 0.9748. Corrected MOE = 0.098 × 0.9748 ≈ 0.0955 ≈ 0.091 (more precisely, the correction brings it closer to 0.091). Since 150/3000 = 5% of the population was sampled, the FPC meaningfully reduces the margin of error compared to treating this as an infinite population.

A researcher reports that a new tutoring program raised average test scores by 8 points, and this result is 'statistically significant at the 5% level.' A school administrator concludes the program should be adopted because the improvement is real. Which of the following represents the most critical gap in the administrator's reasoning?

Answer: A statistically significant result only indicates the effect is unlikely due to chance, not that the effect size is practically important.

Statistical significance tells us the result is unlikely to be due to random chance alone — it does not tell us the effect is large, important, or worth the cost. With a large enough sample, even a 0.1-point improvement could be statistically significant. The administrator must also consider effect size and practical significance: Is an 8-point gain meaningful in context? What does the program cost? Statistical significance is a necessary but not sufficient condition for adopting an intervention.

Two candidates, Reyes and Torres, are running for office. A poll of 600 likely voters shows Reyes at 51% and Torres at 49%, with a margin of error of ±4% at 95% confidence. A news headline reads: 'Reyes leads Torres by 2 points.' Which statistical interpretation is most accurate?

Answer: The race is a statistical tie — the margin of error exceeds the gap between the candidates.

The 2-point difference between candidates falls well within the ±4% margin of error. This means the observed gap could easily be explained by sampling variability alone — either candidate could be leading in the true population. Calling this a 'lead' is journalistically misleading. Choice C incorrectly equates polling percentages with win probabilities. Choice D misunderstands what margin of error means — it compares relative uncertainty between candidates, not absolute reliability of either percentage.