FAST Confidence Intervals and Estimation 2 — Questions and Answers
Question 1: A 95% confidence interval for a population mean is (42, 58). Which statement is correct?
- There is a 95% probability the true mean is between 42 and 58.
- If we repeated sampling many times, about 95% of such intervals would contain the true mean. (Correct answer)
- 95% of individual data values fall between 42 and 58.
- The sample mean has a 95% chance of being between 42 and 58.
Correct answer: If we repeated sampling many times, about 95% of such intervals would contain the true mean.
A confidence interval means that 95% of intervals constructed with this method will capture the true parameter.
Question 2: Increasing the sample size while keeping confidence level constant will:
- Widen the confidence interval.
- Narrow the confidence interval. (Correct answer)
- Have no effect on the interval width.
- Decrease the confidence level.
Correct answer: Narrow the confidence interval.
Larger samples reduce the standard error, which narrows the margin of error and tightens the interval.
Question 3: Which condition is NOT required for constructing a valid confidence interval for a population mean using the z-distribution?
- The population standard deviation is known.
- The sample is randomly selected.
- The population is normally distributed or n is large.
- The sample mean equals the population mean. (Correct answer)
Correct answer: The sample mean equals the population mean.
Sample means rarely equal the population mean exactly; the interval accounts for that uncertainty.
Question 4: A point estimate differs from an interval estimate in that a point estimate:
- Provides a range of plausible values.
- Gives a single value as the best guess for the parameter. (Correct answer)
- Always equals the true population parameter.
- Requires a larger sample size to compute.
Correct answer: Gives a single value as the best guess for the parameter.
A point estimate is a single number used to estimate a population parameter, with no accompanying range.
Question 5: A researcher constructs a 90% confidence interval instead of a 95% confidence interval. Compared to the 95% interval, the 90% interval will be:
- Wider, because lower confidence requires more caution.
- Narrower, because less confidence requires a smaller range. (Correct answer)
- The same width, because sample size did not change.
- Wider, because the z-critical value is larger.
Correct answer: Narrower, because less confidence requires a smaller range.
Lower confidence levels use smaller critical z-values, producing narrower intervals.
Question 6: The margin of error in a confidence interval is defined as:
- The difference between the sample mean and the population mean.
- Half the width of the confidence interval. (Correct answer)
- The standard deviation of the population.
- The probability of a Type I error.
Correct answer: Half the width of the confidence interval.
The margin of error equals the critical value multiplied by the standard error, which is half the total interval width.
Question 7: To reduce the margin of error by half while keeping the confidence level the same, you must:
- Double the sample size.
- Quadruple the sample size. (Correct answer)
- Halve the sample size.
- Increase the confidence level.
Correct answer: Quadruple the sample size.
Since margin of error is proportional to 1/√n, halving it requires multiplying n by 4.
FAST Confidence Intervals and Estimation 2
A 95% confidence interval for a population mean is (42, 58).
Which statement is correct?