Inference from Sample Statistics and Margin of Error Flashcards

A news organization reports a poll result: 'The approval rating for the mayor is 61%, with a margin of error of ±4 percentage points.' Which of the following is not a valid conclusion?

Answer: The approval rating is exactly 61% for the entire population.

A sample statistic is an estimate, not the exact population value; the margin of error indicates that the true value is likely in a range, not pinpointed at 61%.

A pharmaceutical company tests a new drug on a random sample of 900 patients and finds that 73% showed improvement. The margin of error is ±3%. A regulator requires at least 70% improvement rate in the population for approval. Based on the confidence interval, should the drug be approved?

Answer: Yes, because the entire confidence interval (70%–76%) is at or above 70%.

The confidence interval is 70%–76%; since the entire interval meets or exceeds the 70% threshold, the evidence supports approval.

In a confidence interval (p̂ − E, p̂ + E), what does E represent?

Answer: The margin of error

In the standard notation for a confidence interval, E represents the margin of error, the half-width of the interval.

A psychologist samples 169 volunteers and records a mean reaction time of 285 milliseconds with a standard error of 6 ms. Using a critical value of 2, what is the 95% confidence interval?

Answer: (273, 297)

Margin of error = 2 × 6 = 12 ms; 95% CI = 285 ± 12 = (273, 297).

A university estimates from a random sample that 80% of its students are satisfied with campus dining, with a margin of error of ±6%. The administration claims that fewer than 75% of students are actually satisfied. Does the data support this claim?

Answer: No, because 75% is below the lower bound of the confidence interval.

The confidence interval is 74%–86%; since 75% is just inside the interval and the whole range mostly sits above 75%, the data do not support the claim that the true proportion is below 75%.

A survey asks 1,000 randomly selected adults whether they support a policy, and 58% say yes with a margin of error of ±3%. A separate survey of 250 randomly selected adults finds 56% say yes with a margin of error of ±6%. Are the two surveys consistent?

Answer: Yes, because the confidence intervals (55%–61% and 50%–62%) overlap.

The confidence intervals overlap significantly (55%–61% and 50%–62%), so the two estimates are statistically consistent.

A researcher claims that margin of error only depends on sample size. A statistician disagrees. Which factor, besides sample size, also affects margin of error?

Answer: The confidence level chosen

Margin of error depends on both sample size (1/√n) and the confidence level (the critical value z*), so increasing confidence level increases margin of error even with a fixed sample size.