The Pew Research Center interviewed 2300 randomly selected Americans about their attitude about the current economy. Asked "are you in a better financial place than you were 10 years ago?" people either responded "yes" or "no." To calculate the confidence interval for the true percent of Americans who felt they were in a better financial place, we calculated basic statistics for a 95% confidence interval and found the sample proportion of the people who responded "yes" to be 76% with a margin of error of 3%. What is the lower bound of the 95% confidence interval?单项选择题
A
92%
B
73%
C
1748
D
69%

Question

The Pew Research Center interviewed 2300 randomly selected Americans about their attitude about the current economy. Asked "are you in a better financial place than you were 10 years ago?" people either responded "yes" or "no." To calculate the confidence interval for the true percent of Americans who felt they were in a better financial place, we calculated basic statistics for a 95% confidence interval and found the sample proportion of the people who responded "yes" to be 76% with a margin of error of 3%. What is the lower bound of the 95% confidence interval?单项选择题

A

92%

B

73%

C

1748

D

69%

BlackTom AI · Accepted Answer

For a sample proportion p̂ = 76% and a margin of error (ME) = 3%, the lower bound of the 95% confidence interval is p̂ − ME.

1. Relevant knowledge points and concepts
- Confidence interval for a proportion is given by p̂ ± margin of error. The margin of error is already computed (usually z*√[p̂(1−p̂)/n]), so we simply add and subtract it from the sample proportion.
- Percentages can be handled either as percents or decimals; subtracting percentage points (3 percentage points) from 76% yields the lower bound in percentage form.

2. Complete reasoning process step by step
- Given: sample proportion p̂ = 76% and margin of error ME = 3%.
- Lower bound = p̂ − ME.

3. Calculations with intermediate steps
- Convert to decimals if desired: p̂ = 0.76, ME = 0.03.
- Lower bound (decimal) = 0.76 − 0.03 = 0.73.
- Convert back to percent: 0.73 × 100% = 73%.

4. Why the correct answer is correct
- The 95% confidence interval is symmetric around the sample proportion when using a standard margin of error. Subtracting the 3 percentage-point margin from 76% gives 73%, so the lower bound is 73%.

5. Why the other multiple-choice options are incorrect
- 92%: This would be 76% + 16%, not related to the given margin of error; incorrect.
- 1748: This is the raw count of respondents who said “yes” (0.76 × 2300 = 1748). It is a count, not a percentage or confidence-interval bound.
- 69%: This would imply a margin of error of 7 percentage points (76% − 7% = 69%), which contradicts the given ME of 3%.