Inference from sample statistics and margin of error questions ask what a random sample can say about a whole population.
An estimate with a margin of error defines a plausible range: a sample mean of fifty with a margin of three suggests the true mean is likely between forty-seven and fifty-three, though not certainly. Correct choices usually use careful wording such as plausible, while wrong ones claim the true value equals the estimate exactly or that every individual falls in the range.
Larger random samples produce smaller margins of error when the method and confidence level stay the same. Conclusions apply only to the population that was actually sampled.
Sample questions
Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.
Question 1Mid
Two research teams each estimate the mean commute time for workers in the same large metro area. Team A uses a random sample of 80 workers; Team B uses a random sample of 800 workers from the same population. Both teams use the same survey method and the same confidence level. Which of the following is most likely true?
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Answer: B — Team B’s estimate will have a smaller associated margin of error than Team A’s estimate.
For random samples from the same population under comparable methods, a larger sample size generally produces a smaller margin of error. Team B’s sample is ten times as large, so its MoE is expected to be smaller; sharing a population does not make the MoEs equal.
Question 2Easier
Based on a random sample, researchers estimate that the proportion of adults in a county who volunteer at least once a month is 0.28, with an associated margin of error of 0.05. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of adults in the county who volunteer at least once a month?
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Answer: A — It is plausible that the proportion is between 0.23 and 0.33.
The interval 0.28±0.05 is 0.23 to 0.33, so it is plausible that the population proportion lies in that interval. Values below 0.23 or above 0.33 are not supported by the estimate and margin of error, and claiming the proportion equals 0.28 exactly overstates what the estimate and margin of error support.
Question 3Mid
Based on a random sample of tomatoes from a farm, a researcher estimates that the mean weight of tomatoes on the farm is 148 grams, with an associated margin of error of 6 grams. Which of the following is the most appropriate conclusion about the mean weight of tomatoes on the farm?
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Answer: A — It is plausible that the mean weight is between 142 and 154 grams.
The estimate plus or minus the margin of error gives the interval 148−6=142 to 148+6=154. The most appropriate conclusion is that the population mean plausibly lies in that interval; exact equality, values outside the interval, or claims about every individual tomato are not supported.
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