AP Stats Confidence Intervals and Estimation 3 — Questions and Answers
Question 1: A 90% confidence interval for μ is (18.2, 23.8). A student claims the interval means 90% of individuals in the population score between 18.2 and 23.8. This statement is:
- Incorrect; the interval estimates the population mean, not individual values. (Correct answer)
- Correct; the interval captures 90% of data values.
- Correct; this is the definition of a confidence interval.
- Incorrect; it should say 95% of individuals.
Correct answer: Incorrect; the interval estimates the population mean, not individual values.
Confidence intervals estimate population parameters (like μ), not individual data values.
Question 2: A random sample of 50 students has a mean score of 78 with s = 10. The t* critical value for a 95% CI with 49 df is approximately 2.010. What is the margin of error?
- 2.84 (Correct answer)
- 1.41
- 0.28
- 20.10
Correct answer: 2.84
ME = t* × s/√n = 2.010 × 10/√50 = 2.010 × 1.414 ≈ 2.84.
Question 3: Which scenario requires a t-interval rather than a z-interval for estimating a population mean?
- The population standard deviation σ is unknown and must be estimated by s. (Correct answer)
- The sample size is greater than 30.
- The population is known to be normal.
- The margin of error is very small.
Correct answer: The population standard deviation σ is unknown and must be estimated by s.
A t-interval is used when σ is unknown; the t-distribution accounts for the added uncertainty of estimating σ with s.
Question 4: A polling organization wants to estimate a proportion with a margin of error of at most 0.03 at 95% confidence. Using z* = 1.96 and p̂ = 0.5, the minimum sample size is approximately:
- 1068 (Correct answer)
- 534
- 268
- 2135
Correct answer: 1068
n = (z*/ME)² × p̂(1-p̂) = (1.96/0.03)² × 0.25 ≈ 42.67² × 0.25 ≈ 1068.
Question 5: When constructing a confidence interval, the 10% condition (n ≤ 0.10N) is needed to ensure:
- The observations are approximately independent. (Correct answer)
- The sample is representative.
- The distribution is approximately normal.
- The sample size is large enough.
Correct answer: The observations are approximately independent.
The 10% condition ensures that sampling without replacement doesn't violate the independence assumption used in the standard error formula.
Question 6: A confidence interval is constructed from a sample. If the procedure is repeated 200 times with independent samples, approximately how many intervals would be expected to capture the true parameter at 95% confidence?
- 190 (Correct answer)
- 200
- 95
- 100
Correct answer: 190
At 95% confidence, approximately 95% × 200 = 190 intervals would capture the true parameter in the long run.
Question 7: A two-sided 95% confidence interval for μ uses t* = 2.045 (df = 29). To construct a 90% CI with the same data, the new t* would be:
- Smaller than 2.045 (Correct answer)
- Larger than 2.045
- Equal to 2.045
- Equal to 1.96
Correct answer: Smaller than 2.045
Lower confidence level (90% vs 95%) requires a smaller critical value, resulting in a narrower interval.
AP Stats Confidence Intervals and Estimation 3
A 90% confidence interval for μ is (18.2, 23.8).
A student claims the interval means 90% of individuals in the population score between 18.2 and 23.8.
This statement is: