Bluebook SAT Test Inference from Sample Statistics and Margin of Error 5 — Questions and Answers
Question 1: The table below shows results from two surveys about a proposed tax increase: Survey X: 600 respondents, 54% in favor, margin of error ±4% Survey Y: 150 respondents, 58% in favor, margin of error ±8% Which survey provides a more precise estimate of the proportion in favor, and why?
- Survey Y, because a higher percentage supports the tax increase.
- Survey X, because a larger sample produces a smaller margin of error. (Correct answer)
- Survey Y, because it was conducted more recently.
- Both surveys are equally precise since both use random sampling.
Correct answer: Survey X, because a larger sample produces a smaller margin of error.
Survey X uses a larger sample (600 vs. 150), yielding a smaller margin of error (±4% vs. ±8%), making it more precise.
Question 2: A study estimates that the mean annual income of workers in an industry is $52,000, with a 95% confidence interval of ($49,500, $54,500). A labor economist claims the true mean income in that industry is $55,000. Which evaluation is most appropriate?
- The claim is supported because $55,000 is close to the upper bound.
- The claim is not supported because $55,000 falls above the upper bound of the interval. (Correct answer)
- The claim is supported because the sample mean is below $55,000.
- The claim cannot be evaluated from a confidence interval alone.
Correct answer: The claim is not supported because $55,000 falls above the upper bound of the interval.
Since $55,000 is above the upper bound of $54,500, it falls outside the confidence interval and is not a plausible value for the true mean.
Question 3: A polling agency surveys 1,600 likely voters and reports that 49% plan to vote for a candidate, with a margin of error of ±2.5 percentage points. Which conclusion is best supported?
- The candidate will lose because fewer than 50% support her.
- A majority support could exist because the confidence interval extends above 50%. (Correct answer)
- Exactly 49% of all likely voters plan to vote for the candidate.
- The margin of error means the result is not statistically significant.
Correct answer: A majority support could exist because the confidence interval extends above 50%.
The confidence interval is 46.5%–51.5%; since this range includes values above 50%, majority support is a plausible population outcome.
Question 4: A sample of 400 college students finds that 30% own a car. If the margin of error is ±4%, what is the 95% confidence interval, and which value is consistent with the population proportion?
- 26%–34%; a true proportion of 35% is plausible.
- 26%–34%; a true proportion of 31% is plausible. (Correct answer)
- 28%–32%; a true proportion of 27% is not plausible.
- 26%–34%; a true proportion of 25% is plausible.
Correct answer: 26%–34%; a true proportion of 31% is plausible.
The confidence interval is 30% ± 4% = 26%–34%; 31% falls within this range and is therefore a plausible population proportion.
Question 5: To estimate the average price of a gallon of milk in a large city, an economist samples 100 stores and finds a mean of $3.85 with a margin of error of ±$0.20 at 95% confidence. If she wants the margin of error to be ±$0.10, approximately how many stores must she sample?
- 200 stores
- 400 stores (Correct answer)
- 50 stores
- 800 stores
Correct answer: 400 stores
To halve the margin of error, you must quadruple the sample size (since MoE ∝ 1/√n): 100 × 4 = 400 stores.
Question 6: A researcher reports that the 90% confidence interval for mean customer wait time is (4.2 min, 5.8 min). The 95% confidence interval using the same data would be:
- Narrower than (4.2 min, 5.8 min)
- Wider than (4.2 min, 5.8 min) (Correct answer)
- Identical to (4.2 min, 5.8 min)
- Shifted higher than (4.2 min, 5.8 min)
Correct answer: Wider than (4.2 min, 5.8 min)
Increasing the confidence level requires a wider interval to ensure the parameter is captured more often under repeated sampling.
Question 7: A random sample of 500 adults estimates that the mean number of books read per year is 12.4, with a standard error of 0.5. What is the approximate 95% confidence interval, using a critical value of 2?
- (11.4, 13.4) (Correct answer)
- (12.0, 12.8)
- (11.9, 12.9)
- (10.4, 14.4)
Correct answer: (11.4, 13.4)
Margin of error = 2 × 0.5 = 1.0; the 95% confidence interval is 12.4 ± 1.0 = (11.4, 13.4).
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 5
The table below shows results from two surveys about a proposed tax increase:
Survey X: 600 respondents, 54% in favor, margin of error ±4%
Survey Y: 150 respondents, 58% in favor, margin of error ±8%
Which survey provides a more precise estimate of the proportion in favor, and why?