FAST Confidence Intervals and Estimation 3 — Questions and Answers
Question 1: When the population standard deviation is unknown and the sample size is small, which distribution is used to construct a confidence interval for the mean?
- Standard normal (z) distribution.
- Chi-square distribution.
- t-distribution. (Correct answer)
- F-distribution.
Correct answer: t-distribution.
The t-distribution is used when σ is unknown and n is small, because it accounts for additional uncertainty in estimating variability.
Question 2: As degrees of freedom in the t-distribution increase, the t-distribution:
- Becomes more skewed.
- Approaches the standard normal distribution. (Correct answer)
- Widens indefinitely.
- Becomes bimodal.
Correct answer: Approaches the standard normal distribution.
With very large degrees of freedom, the t-distribution's tails shrink and it converges to the z-distribution.
Question 3: A 99% confidence interval is constructed from a sample. Which of the following is a correct interpretation?
- 99% of all possible sample means fall within this interval.
- We are 99% confident that the true population mean lies within this interval. (Correct answer)
- There is a 1% chance the sample was collected incorrectly.
- The population parameter has a 99% probability of being exactly the sample mean.
Correct answer: We are 99% confident that the true population mean lies within this interval.
Confidence is expressed in the long-run reliability of the procedure, not in any single probability statement about the parameter.
Question 4: If a 95% confidence interval for a proportion is (0.40, 0.60), the point estimate of the proportion is:
- 0.40
- 0.60
- 0.50 (Correct answer)
- 0.20
Correct answer: 0.50
The point estimate is the midpoint of the confidence interval: (0.40 + 0.60) / 2 = 0.50.
Question 5: Which of the following would INCREASE the width of a confidence interval?
- Decreasing the confidence level from 99% to 95%.
- Increasing the sample size.
- Decreasing the population variability.
- Increasing the confidence level from 95% to 99%. (Correct answer)
Correct answer: Increasing the confidence level from 95% to 99%.
Higher confidence requires a larger critical value, which increases the margin of error and widens the interval.
Question 6: A sample of 36 scores has a mean of 80 and standard deviation of 12. The standard error of the mean is:
- 12
- 2 (Correct answer)
- 0.33
- 72
Correct answer: 2
Standard error = s / √n = 12 / √36 = 12 / 6 = 2.
Question 7: Which type of estimation provides a range of values likely to contain the true parameter?
- Point estimation
- Interval estimation (Correct answer)
- Parameter estimation
- Regression estimation
Correct answer: Interval estimation
Interval estimation yields a range (lower bound, upper bound) that likely contains the population parameter.
FAST Confidence Intervals and Estimation 3
When the population standard deviation is unknown and the sample size is small, which distribution is used to construct a confidence interval for the mean?