Question 1 Foundation
When should a -test be used instead of a -test?
Answer. When is unknown and the sample is small.
Estimating by adds uncertainty, which the heavier tails of the -distribution account for.
The t-Test
Original questions written at exam standard. Work each one before you open the solution.
Difficulty
Question 2 Foundation
How many degrees of freedom does a one-sample -test with have?
Answer.
. One is lost because the sample mean was computed from the same data.
Question 3 Standard
A sample of 16 has and , testing . Find .
Answer.
SE , so .
Question 4 Standard
Describe how the -distribution differs from the standard normal.
Answer. Same shape but with heavier tails, so larger critical values.
The heavier tails reflect the extra variability from estimating . As the degrees of freedom rise, converges to the normal.
Question 5 Stretch
A 95% confidence interval built from a sample contains the value claimed by . What does a two-tailed test at 5% conclude?
Answer. Do not reject .
The interval and the test are equivalent: a test rejects exactly when the corresponding interval excludes the claimed value.
Question 6 Stretch
Why is a paired test used for before-and-after measurements on the same subjects?
Answer. The two measurements are not independent, and pairing removes between-subject variation.
Testing the differences isolates the treatment effect. Subjects may differ enormously from each other while their individual changes are measured precisely.