Bluebook SAT Test Inference from Sample Statistics and Margin of Error 7 — Questions and Answers
Question 1: A researcher finds that a 95% confidence interval for mean exam scores is (72, 88). A school board member says, 'There's a 95% chance the true mean is in this interval.' Is this statement correct?
- Yes, because confidence intervals are probability statements about the true parameter.
- No, because once an interval is calculated, the true mean is either in it or not — it's the method that has 95% success rate. (Correct answer)
- Yes, but only if the sample was random.
- No, because the true mean equals the sample mean in large samples.
Correct answer: No, because once an interval is calculated, the true mean is either in it or not — it's the method that has 95% success rate.
The 95% refers to the procedure: 95% of all confidence intervals constructed this way contain the true mean; once a specific interval is computed, it either contains the true mean or it does not.
Question 2: A random sample of 400 light bulbs from a manufacturer has a mean lifetime of 1,200 hours with a standard error of 20 hours. The manufacturer claims bulbs last more than 1,250 hours on average. What does the 95% confidence interval (using critical value ≈ 2) suggest about this claim?
- The claim is supported because 1,250 is within 2 standard errors.
- The claim is not supported because 1,250 is above the confidence interval (1,160–1,240). (Correct answer)
- The claim is supported because the sample mean is 1,200.
- The claim is inconclusive without knowing the full population data.
Correct answer: The claim is not supported because 1,250 is above the confidence interval (1,160–1,240).
The 95% CI is 1,200 ± 40 = (1,160, 1,240); since 1,250 is above the upper bound, it is not a plausible population mean, so the manufacturer's claim is not supported.
Question 3: A study of 400 randomly selected households finds a mean monthly water usage of 3,400 gallons with a margin of error of ±150 gallons. The city's target average usage is 3,600 gallons. Is the target consistent with the sample data?
- No, because 3,600 is more than the sample mean.
- Yes, because 3,600 gallons falls within the confidence interval of 3,250–3,550 gallons.
- No, because 3,600 gallons falls above the upper bound of the confidence interval. (Correct answer)
- Yes, because confidence intervals are never used to assess targets.
Correct answer: No, because 3,600 gallons falls above the upper bound of the confidence interval.
The confidence interval is 3,400 ± 150 = (3,250, 3,550); since 3,600 is above the upper bound, it is not plausible given the sample.
Question 4: A survey reports: 'We are 95% confident that between 34% and 42% of adults exercise daily.' What is the sample proportion and margin of error from this report?
- Sample proportion = 34%, margin of error = 8%
- Sample proportion = 38%, margin of error = 4% (Correct answer)
- Sample proportion = 42%, margin of error = 8%
- Sample proportion = 38%, margin of error = 8%
Correct answer: Sample proportion = 38%, margin of error = 4%
Sample proportion = midpoint = (34 + 42)/2 = 38%; margin of error = half-width = (42 − 34)/2 = 4%.
Question 5: Two different analysts use the same random sample of 500 observations to compute confidence intervals, but Analyst A uses 90% confidence while Analyst B uses 99% confidence. Which statement is true?
- Analyst A's interval is wider because lower confidence captures more values.
- Analyst B's interval is wider because higher confidence requires a larger critical value. (Correct answer)
- Both intervals have the same width because the sample is identical.
- Analyst A's interval is more likely to contain the true parameter.
Correct answer: Analyst B's interval is wider because higher confidence requires a larger critical value.
Higher confidence levels use larger critical values (e.g., z* = 2.576 for 99% vs. 1.645 for 90%), resulting in a wider interval to capture the true parameter more reliably.
Question 6: A poll of 625 randomly selected residents shows 48% oppose a new development. The margin of error is ±4%. A city council member says this proves that fewer than half of all residents oppose the development. Is this conclusion valid?
- Yes, because 48% is less than 50%.
- No, because the confidence interval (44%–52%) includes values above 50%, making majority opposition plausible. (Correct answer)
- Yes, because the sample proportion is the best single estimate.
- No, because polls can only be used to show support, not opposition.
Correct answer: No, because the confidence interval (44%–52%) includes values above 50%, making majority opposition plausible.
The confidence interval of 44%–52% includes values above 50%, so majority opposition (>50%) is a plausible population outcome; the council member's conclusion is not supported.
Question 7: A study estimates from a random sample of 100 runners that the mean marathon completion time is 4 hours 12 minutes, with a margin of error of ±18 minutes. Another researcher claims the mean time is under 3 hours 50 minutes. Is this claim plausible?
- Yes, because 3 hours 50 minutes is close to the sample mean.
- No, because 3 hours 50 minutes (230 min) is well below the lower bound of the confidence interval. (Correct answer)
- Yes, because the sample only included 100 runners.
- No, because marathon times are always above 4 hours.
Correct answer: No, because 3 hours 50 minutes (230 min) is well below the lower bound of the confidence interval.
The confidence interval is 252 ± 18 = (234, 270) minutes; 230 minutes (3:50) is below the lower bound of 234, making the claim implausible.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 7
A researcher finds that a 95% confidence interval for mean exam scores is (72, 88).
A school board member says, 'There's a 95% chance the true mean is in this interval.' Is this statement correct?