Bluebook SAT Test Inference from Sample Statistics and Margin of Error 6 — Questions and Answers
Question 1: 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?
- The true approval rating is likely between 57% and 65%.
- The approval rating is exactly 61% for the entire population. (Correct answer)
- A true approval rating of 64% is consistent with the poll results.
- The poll results are consistent with a true approval rating below 65%.
Correct 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%.
Question 2: 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?
- No, because 73% is too close to 70% to be certain.
- Yes, because the entire confidence interval (70%–76%) is at or above 70%. (Correct answer)
- No, because the margin of error makes the true proportion uncertain.
- Yes, because the sample proportion is always the true proportion.
Correct 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.
Question 3: In a confidence interval (p̂ − E, p̂ + E), what does E represent?
- The exact population proportion
- The margin of error (Correct answer)
- The standard deviation of the population
- The sample size needed for precision
Correct 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.
Question 4: 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?
- (273, 297) (Correct answer)
- (279, 291)
- (283, 287)
- (269, 301)
Correct answer: (273, 297)
Margin of error = 2 × 6 = 12 ms; 95% CI = 285 ± 12 = (273, 297).
Question 5: 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?
- Yes, because the lower bound is 74%, which is below 75%.
- No, because the entire confidence interval (74%–86%) includes values well above 75%.
- Yes, because 80% minus the margin of error is near 75%.
- No, because 75% is below the lower bound of the confidence interval. (Correct answer)
Correct 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%.
Question 6: 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?
- No, because the point estimates differ by 2 percentage points.
- Yes, because the confidence intervals (55%–61% and 50%–62%) overlap. (Correct answer)
- No, because surveys of different sizes cannot be compared.
- Yes, but only if both were conducted by the same organization.
Correct 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.
Question 7: 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?
- The mean of the population
- The confidence level chosen (Correct answer)
- Whether the sample is from a rural or urban area
- The number of researchers analyzing the data
Correct 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.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 6
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?