Questions: Z-Tests and T-Tests for Means

Question 1

A researcher tests whether a sample mean differs from a known value. She knows the exact population standard deviation σ. Her colleague advises: 'Always use the t-test — it's more conservative and safer.' Should she follow this advice?

• A. Yes — the t-test's heavier tails provide better protection against false positives in all situations
• B. Yes — t-tests are always preferred at small sample sizes, regardless of what is known
• C. No — when σ is known, the z-test is the exact and correct procedure; using the t-test introduces unnecessary conservatism without justification
• D. No — when σ is known, a chi-squared test should be used instead

Answer: (C)

The t-test's heavier tails exist to account for uncertainty in estimating σ from data. When σ is known exactly, there is no such uncertainty, and the z-test is the correct exact procedure. Using the t-test in this case over-inflates uncertainty without scientific justification — it is more conservative than the data warrants. The choice between z-test and t-test is not a safety preference; it is determined by whether σ is known or estimated.

Question 2

As sample size n increases without bound, what happens to the relationship between the t-test and the z-test?

• A. The t-test becomes increasingly conservative relative to the z-test
• B. The z-test becomes invalid because the Central Limit Theorem assumptions break down
• C. The two tests converge because s converges to σ and the t-distribution approaches the standard normal
• D. The t-test automatically switches to the standard normal distribution when n > 30

Answer: (C)

As n → ∞, the sample standard deviation s is estimated from more data and converges to the true σ. With this convergence, the extra variability that causes the t-distribution's heavier tails disappears. The t-distribution with n−1 degrees of freedom approaches the standard normal as n → ∞. The distinction between t-test and z-test matters primarily at small sample sizes where s is a poor estimator of σ.

Question 3

The t-distribution has heavier tails than the standard normal because estimating σ from data introduces additional variability that must be accounted for.

• T. True
• F. False

Answer: True

When you substitute s for σ in the test statistic, s itself varies from sample to sample — it is a random variable, not a fixed constant. This extra source of variability makes extreme values of the test statistic more likely than under the standard normal. The t-distribution captures this by assigning more probability to its tails. Fewer degrees of freedom (smaller n) means s is estimated from less data, so it varies more, and the tails get heavier.

Question 4

When the population standard deviation σ is known, a t-test with n−1 degrees of freedom is the appropriate procedure.

• T. True
• F. False

Answer: False

When σ is known, the z-test is the correct procedure. The t-test exists specifically to handle the case where σ is unknown and must be estimated by s. Using the t-test when σ is known over-inflates the uncertainty in the test, making it harder to reject a false null hypothesis than the data actually warrants.

Question 5

Why does the t-test use n−1 degrees of freedom rather than n, and how does this make the test more conservative at small sample sizes?

Model answer: After computing the sample mean X̄ from n data points, only n−1 of the deviations (xᵢ − X̄) are free to vary independently — the last deviation is determined by the constraint that all deviations sum to zero. So s is estimated from n−1 independent pieces of information. Fewer degrees of freedom give the t-distribution heavier tails, which means the critical threshold for rejecting H₀ is higher, demanding stronger evidence. At small n, s is highly variable and may be far from σ, so the extra conservatism is appropriate.

The degrees of freedom directly measure how much independent information is used to estimate σ. At n = 2, you have only 1 degree of freedom — the distribution is very wide and demands extreme evidence. As n grows, degrees of freedom increase, the t-distribution tightens toward the normal, and the conservatism shrinks to nothing. The test automatically scales its caution to the amount of information available.