Bluebook SAT Test Inference from Sample Statistics and Margin of Error 1 — Questions and Answers
Question 1: A polling organization surveyed a random sample of 400 registered voters in a state and found that 52% support a ballot measure, with a margin of error of ±4 percentage points at a 95% confidence level. Which of the following is the most appropriate conclusion?
- Exactly 52% of all registered voters in the state support the ballot measure.
- Between 48% and 56% of all registered voters in the state likely support the ballot measure. (Correct answer)
- At least 52% of all registered voters in the state support the ballot measure.
- The ballot measure will pass because more than half the sample supports it.
Correct answer: Between 48% and 56% of all registered voters in the state likely support the ballot measure.
The margin of error creates a confidence interval of 52% ± 4%, meaning the true population proportion is plausibly between 48% and 56%.
Question 2: A researcher increases the sample size in a study from 100 to 900 participants. Assuming all other conditions remain constant, what happens to the margin of error?
- The margin of error triples.
- The margin of error is divided by 3. (Correct answer)
- The margin of error is divided by 9.
- The margin of error remains the same.
Correct answer: The margin of error is divided by 3.
Margin of error is proportional to 1/√n; increasing n from 100 to 900 means √900/√100 = 3, so the margin of error is divided by 3.
Question 3: A survey of 600 randomly selected adults found that 43% have a gym membership, with a margin of error of ±3 percentage points. A researcher claims that fewer than 40% of adults have gym memberships. Based on the survey results, which statement best evaluates this claim?
- The claim is definitely true because the lower bound of the confidence interval is 40%.
- The claim cannot be ruled out because 40% falls just outside the confidence interval.
- The claim is plausible because values below 40% fall just outside the interval, making it uncertain. (Correct answer)
- The claim is supported because the sample proportion is above 43%.
Correct answer: The claim is plausible because values below 40% fall just outside the interval, making it uncertain.
The 95% confidence interval is 40%–46%; since 40% is the lower bound, values below 40% are just outside the plausible range, making the claim unlikely but the result is close enough to warrant caution.
Question 4: Two surveys are conducted to estimate the proportion of students who prefer online learning. Survey A samples 200 students and Survey B samples 800 students. Both use random sampling. Which survey will produce a more precise estimate, and why?
- Survey A, because smaller samples are easier to control for bias.
- Survey B, because larger samples produce smaller margins of error. (Correct answer)
- Both surveys are equally precise because both use random sampling.
- Survey A, because larger samples introduce more variability.
Correct answer: Survey B, because larger samples produce smaller margins of error.
A larger sample size reduces the margin of error (proportional to 1/√n), producing a more precise estimate of the population proportion.
Question 5: A 95% confidence interval for the mean daily screen time of teenagers is reported as (5.8 hours, 6.6 hours). Which of the following is a valid interpretation of this interval?
- 95% of all teenagers have a daily screen time between 5.8 and 6.6 hours.
- There is a 95% probability that the true mean lies between 5.8 and 6.6 hours.
- If the sampling process were repeated many times, approximately 95% of the resulting intervals would contain the true mean. (Correct answer)
- The sample mean is guaranteed to be 6.2 hours.
Correct answer: If the sampling process were repeated many times, approximately 95% of the resulting intervals would contain the true mean.
A confidence interval means that under repeated sampling, 95% of such intervals would capture the true population mean — it does not assign probability to a fixed parameter.
Question 6: A nutritionist randomly samples 250 adults and finds their mean daily calorie intake is 2,050 calories with a margin of error of ±120 calories at a 95% confidence level. The recommended intake is 2,000 calories. Which conclusion is best supported?
- The population mean is exactly 2,050 calories.
- The recommended intake of 2,000 calories is not a plausible population mean.
- The recommended intake of 2,000 calories is a plausible population mean. (Correct answer)
- At least 95% of adults consume more than 2,000 calories.
Correct answer: The recommended intake of 2,000 calories is a plausible population mean.
The confidence interval is 1,930–2,170 calories; since 2,000 falls within this range, it is a plausible value for the true population mean.
Question 7: A city planner surveys a random sample of 500 residents and finds that 68% support a new park, with a margin of error of ±4.5 percentage points. If the planner uses a larger random sample of 2,000 residents, which of the following best describes the expected change?
- The sample proportion will definitely increase.
- The margin of error will decrease to approximately ±2.25 percentage points. (Correct answer)
- The confidence level will automatically increase to 99%.
- The margin of error will decrease to approximately ±1.5 percentage points.
Correct answer: The margin of error will decrease to approximately ±2.25 percentage points.
Quadrupling the sample size (500 to 2,000) reduces the margin of error by a factor of √4 = 2, from ±4.5% to approximately ±2.25%.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 1
A polling organization surveyed a random sample of 400 registered voters in a state and found that 52% support a ballot measure, with a margin of error of ±4 percentage points at a 95% confidence level.
Which of the following is the most appropriate conclusion?