Bluebook SAT Test Inference from Sample Statistics and Margin of Error 10 — Questions and Answers
Question 1: A survey of 400 randomly selected teenagers found that 57% own a smartphone, with a margin of error of ±5 percentage points at 95% confidence. A researcher claims that a majority (more than 50%) of all teenagers own a smartphone. Which statement best evaluates this claim?
- The claim is supported because the entire confidence interval (52% to 62%) is above 50%. (Correct answer)
- The claim is not supported because the margin of error is too large to draw conclusions.
- The claim is uncertain because 50% is within one margin of error of the sample proportion.
- The claim is supported because the sample proportion of 57% exceeds 50%.
Correct answer: The claim is supported because the entire confidence interval (52% to 62%) is above 50%.
The full 95% confidence interval (52% to 62%) lies entirely above 50%, providing strong evidence that a majority owns smartphones.
Question 2: Which of the following actions will NOT change the margin of error in a survey?
- Changing the city in which the survey is conducted while keeping the same sample size and confidence level (Correct answer)
- Increasing the sample size
- Changing the confidence level from 95% to 99%
- Increasing the variability in responses
Correct answer: Changing the city in which the survey is conducted while keeping the same sample size and confidence level
Changing the location of the survey does not affect the margin of error, which depends only on sample size, confidence level, and population variability.
Question 3: A confidence interval for the mean weight of backpacks in a school district is reported as (14.2 lbs, 17.8 lbs) at 95% confidence. A health official claims the true mean backpack weight is 18 lbs. Which conclusion is most appropriate?
- The official's claim is not supported because 18 lbs is outside the confidence interval. (Correct answer)
- The official's claim is supported because 18 lbs is close to the upper bound.
- The official's claim cannot be evaluated without the sample size.
- The official's claim is supported because the interval includes values near 18 lbs.
Correct answer: The official's claim is not supported because 18 lbs is outside the confidence interval.
Since 18 lbs falls outside the 95% confidence interval (14.2, 17.8), the data do not support the official's claim.
Question 4: A random sample of 625 college students estimates that the mean hours of sleep per night is 6.8 hours with a margin of error of ±0.4 hours. What sample size would be needed to reduce the margin of error to ±0.2 hours, assuming the same variability?
- 2,500 (Correct answer)
- 1,250
- 1,875
- 312
Correct answer: 2,500
Halving the margin of error requires quadrupling the sample size: 625 × 4 = 2,500.
Question 5: A polling firm reports that 46% of likely voters support a candidate, with a 95% confidence interval of (41%, 51%). A campaign manager concludes that the candidate is winning because 46% is the sample proportion. Which statement best evaluates this conclusion?
- The conclusion is not valid because the confidence interval includes values below 50%. (Correct answer)
- The conclusion is valid because 46% is the best estimate of the true proportion.
- The conclusion is valid because the confidence interval is centered near 50%.
- The conclusion is not valid because the sample size is likely too small.
Correct answer: The conclusion is not valid because the confidence interval includes values below 50%.
Because the 95% confidence interval (41%–51%) includes values below 50%, it is plausible that fewer than half of voters support the candidate.
Question 6: A manufacturer tests a random sample of 100 light bulbs and finds a mean lifespan of 1,200 hours with a margin of error of ±40 hours at 95% confidence. A customer claims the true mean lifespan is at least 1,250 hours. Which assessment is most appropriate?
- The claim is not supported because 1,250 hours is outside the confidence interval. (Correct answer)
- The claim is supported because 1,250 hours is only slightly above the upper bound.
- The claim is supported because the sample mean is high enough.
- The claim cannot be assessed without knowing the standard deviation.
Correct answer: The claim is not supported because 1,250 hours is outside the confidence interval.
The 95% CI is (1,160, 1,240); since 1,250 lies above this interval, the data do not support the claim of at least 1,250 hours.
Question 7: A researcher computes a 90% confidence interval and a 99% confidence interval from the same dataset. How do these intervals compare?
- The 99% confidence interval is wider than the 90% confidence interval. (Correct answer)
- The 90% confidence interval is wider than the 99% confidence interval.
- Both intervals have the same width because they use the same data.
- The 99% confidence interval is narrower because it is more precise.
Correct answer: The 99% confidence interval is wider than the 90% confidence interval.
A higher confidence level requires a wider interval to capture the true parameter with greater certainty.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 10
A survey of 400 randomly selected teenagers found that 57% own a smartphone, with a margin of error of ±5 percentage points at 95% confidence.
A researcher claims that a majority (more than 50%) of all teenagers own a smartphone.
Which statement best evaluates this claim?