AP Stats Confidence Intervals and Estimation 2 โ Questions and Answers
Question 1: A 95% confidence interval for a population mean is (42.1, 47.9). Which interpretation is correct?
- We are 95% confident the population mean falls between 42.1 and 47.9. (Correct answer)
- There is a 95% probability the population mean is between 42.1 and 47.9.
- 95% of all sample means fall between 42.1 and 47.9.
- The population mean equals 45 with 95% accuracy.
Correct answer: We are 95% confident the population mean falls between 42.1 and 47.9.
A confidence interval means we are 95% confident (in the procedure) that the interval captures the true population mean.
Question 2: Which condition must be verified before constructing a one-sample t-interval for a mean when n = 15?
- The population distribution is approximately normal or the data show no strong skew/outliers. (Correct answer)
- The sample size is at least 30.
- The population standard deviation is known.
- The sample was taken from at least 10% of the population.
Correct answer: The population distribution is approximately normal or the data show no strong skew/outliers.
For small samples (n < 30), the nearly normal condition must be checked via a dotplot or boxplot.
Question 3: A researcher uses a 99% confidence level instead of 95% for the same data. What happens to the interval?
- It becomes wider. (Correct answer)
- It becomes narrower.
- It stays the same width but shifts location.
- It becomes wider only if n is large.
Correct answer: It becomes wider.
Higher confidence requires a larger critical value (z* or t*), which increases the margin of error and widens the interval.
Question 4: The margin of error in a confidence interval for a proportion is 0.04. If the sample size is quadrupled, the new margin of error is approximately:
- 0.02 (Correct answer)
- 0.01
- 0.08
- 0.04
Correct answer: 0.02
Margin of error is proportional to 1/โn, so quadrupling n cuts the margin of error in half.
Question 5: A confidence interval for a proportion is calculated as (0.58, 0.72). What is the sample proportion pฬ?
- 0.65 (Correct answer)
- 0.58
- 0.72
- 0.14
Correct answer: 0.65
The sample proportion is the midpoint of the interval: (0.58 + 0.72)/2 = 0.65.
Question 6: Which of the following would NOT reduce the margin of error in a confidence interval?
- Increasing the confidence level from 95% to 99%. (Correct answer)
- Increasing the sample size.
- Using a smaller population standard deviation.
- Decreasing the confidence level from 95% to 90%.
Correct answer: Increasing the confidence level from 95% to 99%.
Increasing the confidence level raises the critical value, which increases (not decreases) the margin of error.
Question 7: For a one-sample z-interval for a proportion, the standard error is calculated using which formula?
- โ(pฬ(1-pฬ)/n) (Correct answer)
- ฯ/โn
- s/โn
- โ(p(1-p))
Correct answer: โ(pฬ(1-pฬ)/n)
The standard error for a sample proportion uses the sample proportion pฬ in place of the unknown population proportion p.
AP Stats Confidence Intervals and Estimation 2
A 95% confidence interval for a population mean is (42.1, 47.9).
Which interpretation is correct?