Bluebook SAT Test Inference from Sample Statistics and Margin of Error 9 — Questions and Answers
Question 1: A random sample of 500 households in a county found that 38% have solar panels installed, with a margin of error of ±4 percentage points at 95% confidence. Which of the following intervals represents the plausible range for the true proportion of all county households with solar panels?
- 34% to 42% (Correct answer)
- 36% to 40%
- 30% to 46%
- 38% to 42%
Correct answer: 34% to 42%
The confidence interval is formed by adding and subtracting the margin of error from the sample proportion: 38% ± 4% gives 34% to 42%.
Question 2: A research team surveys 900 randomly selected adults and finds that 61% support a proposed policy. The team then conducts a second survey using a random sample of 225 adults. Assuming the true proportion stays the same, how does the margin of error for the second survey compare to the first?
- It doubles (Correct answer)
- It is four times as large
- It is halved
- It stays the same
Correct answer: It doubles
Margin of error is proportional to 1/√n; reducing n from 900 to 225 (dividing by 4) doubles 1/√n, so the margin of error doubles.
Question 3: A 95% confidence interval for the mean hours of weekly exercise among adults in a city is (3.1 hours, 4.5 hours). A fitness company claims the true mean is 5 hours per week. Which statement best evaluates this claim?
- The claim is inconsistent with the data because 5 hours is outside the confidence interval. (Correct answer)
- The claim is consistent with the data because 5 hours is close to the interval.
- The claim cannot be evaluated without knowing the sample size.
- The claim is consistent because 95% confidence intervals always include the true mean.
Correct answer: The claim is inconsistent with the data because 5 hours is outside the confidence interval.
Because 5 hours falls outside the 95% confidence interval of (3.1, 4.5), the data do not support the company's claim.
Question 4: A school principal wants to estimate the proportion of students who walk to school. She surveys a random sample of 50 students and obtains a margin of error of ±7%. If she wants to reduce the margin of error to ±3.5%, approximately how large must the new sample be?
- 200 (Correct answer)
- 100
- 150
- 400
Correct answer: 200
Halving the margin of error requires quadrupling the sample size; 50 × 4 = 200.
Question 5: In a study of 800 randomly chosen shoppers, 44% said they prefer buying groceries online. The reported margin of error is ±3.5 percentage points at 95% confidence. A store manager claims that fewer than 40% of shoppers prefer online grocery shopping. Which statement best assesses this claim?
- The claim is plausible because 40% falls just inside the lower boundary of the confidence interval. (Correct answer)
- The claim is definitively false because the sample proportion is above 40%.
- The claim cannot be assessed without knowing the population size.
- The claim is supported because the margin of error is larger than the difference.
Correct answer: The claim is plausible because 40% falls just inside the lower boundary of the confidence interval.
The 95% CI is approximately 40.5% to 47.5%, so 40% barely falls outside it, making the manager's claim not well-supported by the data.
Question 6: A 95% confidence interval for a population proportion is reported as (0.52, 0.68). What is the sample proportion and the margin of error?
- Sample proportion = 0.60; margin of error = 0.08 (Correct answer)
- Sample proportion = 0.52; margin of error = 0.16
- Sample proportion = 0.68; margin of error = 0.08
- Sample proportion = 0.60; margin of error = 0.16
Correct answer: Sample proportion = 0.60; margin of error = 0.08
The sample proportion is the midpoint: (0.52 + 0.68)/2 = 0.60, and the margin of error is half the width: (0.68 − 0.52)/2 = 0.08.
Question 7: A researcher surveys a random sample of 1,024 adults and finds that 30% report feeling stressed daily, with a margin of error of ±3 percentage points at 95% confidence. If the researcher increases the sample to 4,096 adults, what would the new margin of error be, assuming the same proportion?
- ±1.5 percentage points (Correct answer)
- ±6 percentage points
- ±3 percentage points
- ±0.75 percentage points
Correct answer: ±1.5 percentage points
Quadrupling the sample size halves the margin of error: 3% ÷ 2 = 1.5 percentage points.
Bluebook SAT Test Inference from Sample Statistics and Margin of Error 9
A random sample of 500 households in a county found that 38% have solar panels installed, with a margin of error of ±4 percentage points at 95% confidence.
Which of the following intervals represents the plausible range for the true proportion of all county households with solar panels?