Bluebook SAT Test Inference from Sample Statistics and Margin of Error 8 — Questions and Answers
Question 1: A study finds that a 99% confidence interval for the mean is wider than the corresponding 95% confidence interval. A student asks: 'Does the wider interval mean we are less certain about the true mean?' What is the correct response?
- Yes, because a wider interval captures less information.
- No, because a wider interval provides more certainty that the true mean is captured. (Correct answer)
- Yes, because the sample size must have been smaller for the 99% interval.
- No, because both intervals have the same margin of error.
Correct answer: No, because a wider interval provides more certainty that the true mean is captured.
A wider confidence interval actually provides greater certainty of capturing the true mean — the trade-off is precision for confidence.
Question 2: A hospital records the blood pressure of 900 randomly selected patients and finds a mean systolic pressure of 118 mmHg with a margin of error of ±2 mmHg at 95% confidence. A health organization defines hypertension risk as a mean above 120 mmHg. Is the organization's threshold consistent with this sample?
- Yes, because 120 mmHg is within the confidence interval of 116–120 mmHg.
- No, because 120 mmHg is at the upper boundary and values above it fall outside the confidence interval. (Correct answer)
- Yes, because the sample mean of 118 is close enough to 120 to be equivalent.
- No, because confidence intervals cannot be used to evaluate health thresholds.
Correct answer: No, because 120 mmHg is at the upper boundary and values above it fall outside the confidence interval.
The confidence interval is 116–120 mmHg; 120 is the upper boundary, and values above the threshold fall outside the interval, making the population mean exceeding 120 implausible.
Question 3: A researcher wants to estimate the mean height of adult men in a country to within ±0.5 cm at 95% confidence. The standard deviation is estimated at 7 cm. Using n = (z* × σ / E)² with z* = 2, approximately how large a sample is needed?
- 196 men
- 784 men (Correct answer)
- 49 men
- 2,401 men
Correct answer: 784 men
n = (2 × 7 / 0.5)² = (28)² = 784 men.
Question 4: A supermarket surveys a random sample of 900 customers and finds a mean spending of $87 per visit with a standard error of $2. Which statement about the 95% confidence interval is accurate, using z* = 2?
- The confidence interval is ($85, $89), meaning 95% of customers spend in this range.
- The confidence interval is ($83, $91), giving a plausible range for the true mean spending per visit. (Correct answer)
- The confidence interval is ($83, $91), and 95% of individual customers spend in this range.
- The confidence interval is ($85, $89), and the true mean is guaranteed to be $87.
Correct answer: The confidence interval is ($83, $91), giving a plausible range for the true mean spending per visit.
The 95% CI is $87 ± 2($2) = $87 ± $4 = ($83, $91), and it estimates the plausible range for the population mean — not the range of individual customer spending.
Question 5: A confidence interval for mean study hours is (1.8, 3.2). A professor claims students study less than 2 hours on average. Based on the confidence interval, which statement is most accurate?
- The professor's claim is definitely false because the midpoint is 2.5 hours.
- The professor's claim is plausible because values below 2 hours are within the confidence interval. (Correct answer)
- The professor's claim is not plausible because 2 hours is above the lower bound of the interval.
- The professor's claim is true because 1.8 is in the interval.
Correct answer: The professor's claim is plausible because values below 2 hours are within the confidence interval.
Values below 2 hours (down to 1.8) are within the confidence interval, so a true mean below 2 hours is plausible based on the data.
Question 6: A random sample of 50 employees has a mean commute time of 35 minutes. The margin of error at 95% confidence is ±6 minutes. If the sample size increases to 200, what is the new margin of error?
- ±12 minutes
- ±3 minutes (Correct answer)
- ±1.5 minutes
- ±24 minutes
Correct answer: ±3 minutes
Increasing sample size by factor of 4 (50 to 200) reduces margin of error by √4 = 2: new margin of error = 6/2 = ±3 minutes.
Question 7: A random sample of 100 trees in a forest finds a mean height of 42 feet with a 95% confidence interval of (39, 45). A forest manager claims the true mean is 46 feet. Which evaluation is correct?
- The claim of 46 feet is plausible because it is close to the upper bound.
- The claim of 46 feet is not plausible because it falls above the upper bound of the confidence interval. (Correct answer)
- The claim is plausible because the sample size of 100 is too small to rule out 46 feet.
- The claim is plausible because the margin of error makes all values near 45 possible.
Correct answer: The claim of 46 feet is not plausible because it falls above the upper bound of the confidence interval.
46 feet falls above the upper bound of 45 feet, placing it outside the confidence interval and making it an implausible value for the true mean.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 8
A study finds that a 99% confidence interval for the mean is wider than the corresponding 95% confidence interval.
A student asks: 'Does the wider interval mean we are less certain about the true mean?' What is the correct response?