Bluebook SAT Test Inference from Sample Statistics and Margin of Error 3 — Questions and Answers
Question 1: A researcher wants to estimate the mean weight of a species of fish in a lake. Sample 1 consists of 50 randomly caught fish and sample 2 consists of 200 randomly caught fish. Both samples have similar means. Which sample will produce a confidence interval with a smaller width?
- Sample 1, because smaller samples have less variability.
- Sample 2, because larger samples produce narrower confidence intervals. (Correct answer)
- Both samples will produce intervals of equal width.
- Sample 1, because it is easier to ensure a truly random sample.
Correct answer: Sample 2, because larger samples produce narrower confidence intervals.
Larger samples reduce the standard error, resulting in narrower confidence intervals and more precise estimates.
Question 2: The results of a study report: 'We are 95% confident that the mean number of hours adults spend reading per week is between 3.2 and 4.8 hours.' What is the margin of error?
- 0.8 hours (Correct answer)
- 1.6 hours
- 4.0 hours
- 3.2 hours
Correct answer: 0.8 hours
The margin of error is half the confidence interval width: (4.8 − 3.2)/2 = 0.8 hours.
Question 3: A random sample of 64 students shows a mean study time of 3.5 hours per day with a standard deviation of 1.2 hours. If the sample size is increased to 256 students with the same standard deviation, by what factor does the margin of error change?
- It is multiplied by 4.
- It is divided by 4.
- It is multiplied by 2.
- It is divided by 2. (Correct answer)
Correct answer: It is divided by 2.
Sample size increases by a factor of 4 (64 to 256), so √4 = 2; the margin of error is divided by 2.
Question 4: A television network surveys 1,000 randomly selected viewers and reports that 63% watch a particular show, with a margin of error of ±3 percentage points at 95% confidence. Which of the following populations can this estimate validly be generalized to?
- All viewers of all television networks worldwide
- All viewers in the population from which the random sample was drawn (Correct answer)
- All 1,000 viewers who were surveyed
- All adults who own a television
Correct answer: All viewers in the population from which the random sample was drawn
Inferences from a random sample can only be generalized to the population from which that sample was drawn, not to broader populations not represented.
Question 5: A poll estimates that 48% of voters support a candidate, with a margin of error of ±4 percentage points. A competitor's poll estimates 51% support with a margin of error of ±4 percentage points. Which conclusion is most appropriate?
- The second poll is more accurate because it shows higher support.
- The two polls are contradictory and both should be discarded.
- The results are consistent because the confidence intervals overlap. (Correct answer)
- The candidate is guaranteed to lose because neither poll shows clear majority support.
Correct answer: The results are consistent because the confidence intervals overlap.
Poll 1 gives 44%–52% and Poll 2 gives 47%–55%; these intervals overlap, so both results are consistent with a range of true values.
Question 6: A company surveys 400 customers and finds a mean satisfaction score of 7.4 out of 10, with a margin of error of ±0.3. A manager claims the true mean satisfaction score is at most 7.0. Is this claim consistent with the survey?
- Yes, because 7.0 is within one standard deviation of the mean.
- No, because 7.0 is below the lower bound of the confidence interval. (Correct answer)
- Yes, because the company may have surveyed unsatisfied customers.
- No, because the sample size of 400 is too small to trust.
Correct answer: No, because 7.0 is below the lower bound of the confidence interval.
The confidence interval is 7.1 to 7.7; since 7.0 is below the lower bound of 7.1, it is not a plausible value for the true mean.
Question 7: A study uses a random sample to estimate the proportion of adults in a county who exercise regularly. The 95% confidence interval is (0.34, 0.46). Which of the following must be true?
- Exactly 40% of adults in the county exercise regularly.
- The sample proportion is 0.40. (Correct answer)
- At least 95% of county adults exercise between 34% and 46% of the time.
- There is a 95% probability that the true proportion is exactly 0.40.
Correct answer: The sample proportion is 0.40.
The sample proportion is the midpoint of the confidence interval: (0.34 + 0.46)/2 = 0.40.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 3
A researcher wants to estimate the mean weight of a species of fish in a lake.
Sample 1 consists of 50 randomly caught fish and sample 2 consists of 200 randomly caught fish.
Both samples have similar means.
Which sample will produce a confidence interval with a smaller width?