After computing a 95% CI from her data, a statistician reports: 'There is a 95% probability that the true parameter μ lies between 2.1 and 4.7.' What is wrong with this statement?
ANothing — a 95% CI is defined as an interval that contains the true parameter with probability 0.95
BOnce the data are observed and the interval [2.1, 4.7] is realized, μ is fixed and either is or is not in the interval — the 95% describes the procedure across repeated experiments, not this specific realized interval
CThe statement should say 'at least 95% probability' because the coverage guarantee is a lower bound
DThe CI should have been expressed as a probability about the estimator, not the parameter
This is the most common misinterpretation of confidence intervals. In the frequentist framework, θ is a fixed (unknown) constant — it has no probability distribution. Once the interval is computed from observed data, it either contains θ (probability 1) or it doesn't (probability 0). The 95% is a property of the *procedure* L(X), U(X): if we repeated the experiment infinitely many times and computed CIs each time, 95% of those intervals would contain θ. The realized interval [2.1, 4.7] is one draw from that procedure.
Question 2 Multiple Choice
Using the test inversion principle, a statistician finds that the data fail to reject H₀: θ = 4.0 but do reject H₀: θ = 3.5 and H₀: θ = 6.2, all at the 5% level. What follows about the 95% confidence interval?
AThe CI is [3.5, 6.2] — it spans from the boundary of rejection to rejection
Bθ = 4.0 lies inside the 95% CI; θ = 3.5 and θ = 6.2 lie outside (or on the boundary)
CThe CI cannot be determined from test decisions alone; it must be computed directly from the data
DThe CI is a single point at θ = 4.0 because only that value is not rejected
Test inversion is exact: the (1−α) CI is precisely the set C(X) = {θ₀ : H₀: θ = θ₀ is not rejected at level α}. If θ = 4.0 is not rejected but θ = 3.5 and θ = 6.2 are, then 4.0 ∈ C(X) and 3.5, 6.2 ∉ C(X). This is the direct connection between hypothesis testing and confidence intervals — they are two sides of the same inference, not separate procedures.
Question 3 True / False
A confidence interval [L(X), U(X)] is a random interval because L and U are functions of the random data X; the parameter θ is fixed and unknown.
TTrue
FFalse
Answer: True
This is the essential conceptual point. L(X) and U(X) are statistics — they vary from sample to sample. The parameter θ does not move. The probability statement P(L(X) ≤ θ ≤ U(X)) = 1−α describes the distribution of the *interval* across hypothetical repetitions of the experiment, with θ held fixed. Confusing which element is random (the interval, not the parameter) leads to the common misinterpretation that any particular CI has a 95% chance of containing θ.
Question 4 True / False
When a statistician says a 95% CI 'covers' the true parameter, she means that the interval would contain the true θ for 95% of most possible true parameter values.
TTrue
FFalse
Answer: False
Coverage probability is not a statement across parameter values — it is a statement across repeated experiments. Specifically, P_θ(L(X) ≤ θ ≤ U(X)) ≥ 1−α must hold for *every* θ, not just 95% of them. 'Coverage' means: if the true parameter is θ (whatever it is), and you repeat the sampling procedure many times, at least 95% of the resulting intervals will contain θ. The probability is over the randomness in X, not over θ.
Question 5 Short Answer
Explain the test inversion principle: how does inverting a level-α hypothesis test produce a valid (1−α) confidence set? Why does the validity of the CI follow directly from the level guarantee of the test?
Think about your answer, then reveal below.
Model answer: For each candidate parameter value θ₀, run the level-α test of H₀: θ = θ₀ with acceptance region A(θ₀). Define the confidence set C(X) = {θ₀ : X ∈ A(θ₀)} — all values not rejected by the data. Then P_θ(θ ∈ C(X)) = P_θ(X ∈ A(θ)) = 1 − α, because A(θ) is exactly the set of data outcomes where H₀: θ = θ is accepted, which happens with probability 1−α by the level guarantee. The CI validity follows directly: no new derivation is needed beyond the test's level property.
This makes the CI-test duality exact rather than merely analogical. The acceptance region A(θ₀) and the confidence set C(X) are transposes of each other: one is a set of data values for fixed θ₀, the other is a set of parameter values for fixed X. When the underlying test is UMP (uniformly most powerful), the inversion produces the shortest possible CI, connecting test optimality directly to estimation efficiency.