Bluebook SAT Test Inference from Sample Statistics and Margin of Error 2 — Questions and Answers
Question 1: A school district reports that a random sample of 350 students had a mean math score of 74 with a margin of error of ±5 points. Which of the following best describes the plausible range for the true mean score of all students in the district?
- Exactly 74 points
- Between 69 and 79 points (Correct answer)
- Between 74 and 79 points
- Less than 74 points
Correct answer: Between 69 and 79 points
The margin of error of ±5 is subtracted and added to the sample mean of 74, giving a plausible range of 69 to 79.
Question 2: A sample of 100 households shows an average electricity bill of $145 per month with a margin of error of ±$12. A researcher claims the true average is less than $130. Is this claim consistent with the survey results?
- Yes, because $130 is close to the lower bound of the interval.
- No, because $130 is below the confidence interval of $133–$157. (Correct answer)
- Yes, because the sample mean is greater than $130.
- No, because the sample mean equals the true mean.
Correct answer: No, because $130 is below the confidence interval of $133–$157.
The confidence interval runs from $133 to $157; since $130 falls below the lower bound, it is not a plausible value for the population mean.
Question 3: A random sample of 1,600 voters shows 55% prefer Candidate A, with a margin of error of ±2.5%. Another survey of 400 voters shows 57% prefer Candidate A, with a margin of error of ±5%. Which statement is best supported?
- Candidate A is definitely winning because both samples show more than 50% support.
- The larger sample's estimate is more reliable because it has a narrower margin of error. (Correct answer)
- The two surveys contradict each other because their point estimates differ.
- Candidate A's true support is exactly 56%.
Correct answer: The larger sample's estimate is more reliable because it has a narrower margin of error.
A larger sample produces a narrower margin of error, making the estimate from 1,600 voters (±2.5%) more precise and reliable than the estimate from 400 voters (±5%).
Question 4: A health survey of 900 adults estimates that 22% are classified as having high blood pressure, with a margin of error of ±3 percentage points at 95% confidence. What is the range of values consistent with the survey at this confidence level?
- 19% to 22%
- 22% to 25%
- 19% to 25% (Correct answer)
- 16% to 28%
Correct answer: 19% to 25%
The confidence interval is 22% ± 3%, giving a range from 19% to 25%.
Question 5: Which change to a study design would most directly decrease the margin of error in estimating a population proportion?
- Increasing the confidence level from 90% to 99%
- Decreasing the sample size from 500 to 200
- Increasing the sample size from 200 to 800 (Correct answer)
- Using a convenience sample instead of a random sample
Correct answer: Increasing the sample size from 200 to 800
Increasing sample size directly decreases the margin of error; the other options either increase it, maintain it, or introduce bias without improving precision.
Question 6: A 95% confidence interval for the proportion of customers satisfied with a product is (0.62, 0.78). What is the sample proportion and the margin of error?
- Sample proportion = 0.70, margin of error = 0.08 (Correct answer)
- Sample proportion = 0.62, margin of error = 0.16
- Sample proportion = 0.78, margin of error = 0.08
- Sample proportion = 0.70, margin of error = 0.16
Correct answer: Sample proportion = 0.70, margin of error = 0.08
The sample proportion is the midpoint: (0.62 + 0.78)/2 = 0.70, and the margin of error is half the width: (0.78 − 0.62)/2 = 0.08.
Question 7: A 95% confidence interval for the mean commute time in a city is (28 min, 36 min). A city official claims the mean commute time is over 38 minutes. Which statement is most accurate?
- The claim is supported because 38 is near the upper bound.
- The claim is not supported because 38 falls outside the confidence interval. (Correct answer)
- The claim is supported because the sample mean is 32 minutes.
- The claim cannot be evaluated without knowing the exact sample size.
Correct answer: The claim is not supported because 38 falls outside the confidence interval.
Since 38 minutes is outside the confidence interval (28–36 minutes), it is not a plausible value for the true mean, so the official's claim is not supported.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 2
A school district reports that a random sample of 350 students had a mean math score of 74 with a margin of error of ±5 points.
Which of the following best describes the plausible range for the true mean score of all students in the district?