Question 1 Foundation
State the formula for the standard error of the mean.
Answer.
It is the standard deviation of the sampling distribution of , and it shrinks as grows.
Sampling Distributions and Estimation
Original questions written at exam standard. Work each one before you open the solution.
Difficulty
Question 2 Foundation
A population has . Find the standard error for samples of size 100.
Answer.
.
Question 3 Standard
State the value used for a 99% confidence interval.
Answer.
It leaves 0.5% in each tail. Compare 1.96 for 95% and 1.645 for 90%.
Question 4 Standard
Give an unbiased estimator of a population variance from a sample.
Answer.
Dividing by would underestimate systematically, because deviations are measured from the sample mean rather than the true mean.
Question 5 Stretch
A 95% confidence interval is . What happens to it if the sample size is quadrupled, all else equal?
Answer. It halves in width, becoming .
The standard error is divided by , so the margin of error halves from 2 to 1.
Question 6 Stretch
Why is "there is a 95% probability that lies in this interval" incorrect?
Answer. is fixed, not random — it either is or is not in the interval.
The 95% describes the long-run success rate of the PROCEDURE: about 95% of intervals built this way would capture .