Inference from Sample Statistics and Margin of Error Flashcards

A polling organization surveys 600 randomly selected voters and finds that 54% support a ballot measure, with a margin of error of ±4% at a 95% confidence level. A rival poll surveys 150 voters and finds 58% support with a margin of error of ±8%. Which statement best describes what can be concluded from both polls combined?

Answer: The true population proportion almost certainly lies between 50% and 66%, but the overlapping confidence intervals prevent a definitive conclusion about majority support.

Each poll produces a confidence interval: the first gives roughly 50%–58%, the second gives roughly 50%–66%. Both intervals include values at or below 50%, so neither poll alone—nor both together—can definitively confirm majority support. The overlapping intervals are consistent with each other (not contradictory), and the correct takeaway is that the true proportion plausibly spans both ranges. Option B ignores that 50% falls within both intervals. Option C is partially correct in spirit but incorrectly discards relevant data. Option D misunderstands what overlapping or differing point estimates mean.

A researcher wants to estimate the mean number of hours per week teenagers spend on social media. A pilot study of 25 teenagers yields a sample standard deviation of 5 hours. If the researcher wants a margin of error no greater than 1 hour at a 95% confidence level (z* ≈ 1.96), what is the minimum sample size needed?

Answer: 97

The margin of error formula is ME = z*(s/√n). Setting 1 = 1.96 × (5/√n) and solving: √n = 1.96 × 5 / 1 = 9.8, so n = 9.8² = 96.04. Rounding up gives a minimum of 97. Answer B (25) is the pilot size and does not meet the requirement. Answer C (196) would result from forgetting to divide by the margin of error before squaring (using (1.96 × 5)² directly). Answer D (10) is far too small.

Two independent random samples are drawn from the same population. Sample A has n = 400 and produces a 95% confidence interval of (0.43, 0.57) for a proportion. Sample B has n = 100 and produces a 95% confidence interval of (0.38, 0.62). Which of the following is a valid statistical explanation for why Sample B's interval is wider?

Answer: A smaller sample size increases the standard error of the sample proportion, which increases the margin of error.

The margin of error for a proportion is z*√(p̂(1−p̂)/n). With a smaller n, the standard error √(p̂(1−p̂)/n) is larger, which directly widens the confidence interval. Both samples use a 95% confidence level (same z*), eliminating option B. Option C is incorrect because the width of a confidence interval does not depend on how close p̂ is to 0.5 in the way described—and the interval is symmetric around p̂. Option D confuses a larger variance with bias; random sampling from the same population does not 'invalidate' an interval.

A survey reports: 'We are 95% confident that between 41% and 49% of adults in the city favor the new transit policy.' A city council member concludes, 'There is a 95% chance that the true proportion falls in this interval.' Why is the council member's interpretation statistically incorrect?

Answer: The true population proportion is a fixed (non-random) value, so it either is or is not in the interval; the 95% refers to the long-run proportion of such intervals that capture the true value.

A confidence interval is a procedure, not a probability statement about a fixed parameter. The true proportion p is a fixed number—it does not randomly fall in or out of intervals. What 95% confidence means is that if this sampling procedure were repeated many times, 95% of the resulting intervals would contain the true parameter. Once a specific interval is computed, p is either in it or not. The council member's phrasing implies p is random, which is incorrect. Options C and D misidentify what the interval describes.

A study estimates that the average commute time in a city is 34 minutes, with a 90% confidence interval of (31, 37) minutes. A second study of a different city produces a 95% confidence interval of (31, 37) minutes using the same sample size. What must be true about the second city's sample data compared to the first?

Answer: The second city's sample had a smaller standard deviation, since a higher confidence level with the same sample size would otherwise produce a wider interval.

The width of a confidence interval is determined by z*(s/√n). For the same sample size n, a higher confidence level means a larger critical value z* (1.96 vs 1.645). To keep the total interval width the same despite a larger z*, the standard deviation s must be smaller. Option B reverses this logic. Option C is an unsupported inference—identical intervals can arise from very different distributions. Option D contradicts the premise that the same sample size was used.

A quality control inspector samples 200 bolts from a production line and finds that 12 are defective, yielding a 95% confidence interval for the defect rate of approximately (0.031, 0.089). The plant manager claims the true defect rate is 3%. Based on the confidence interval, which of the following is the most statistically precise conclusion?

Answer: The claim of a 3.1% defect rate is at the boundary of the confidence interval, making it marginally plausible but not strongly supported by the data.

The lower bound of the interval is approximately 3.1% (0.031). A claimed defect rate of exactly 3% falls just outside the 95% confidence interval, meaning it is not among the plausible values at this confidence level—but it is extremely close to the boundary. This makes it marginally implausible, not definitively refuted. Option B understates the statistical finding (3% is outside the interval, not 'close enough' to be simply plausible). Option C overstates the conclusion—a sample proportion differing from a claim does not by itself 'definitively refute' anything. Option D is factually wrong: 3% is below the lower bound, meaning the interval does NOT confirm it.