A researcher computes a 95% confidence interval for a population mean and obtains [2.3, 4.7]. She states: 'There is a 95% probability that the true mean lies between 2.3 and 4.7.' What is wrong with this interpretation?
ANothing — this is precisely what a 95% confidence interval means
BShe should say 90%, not 95%, because the interval is symmetric
COnce the data are observed, the interval is fixed and the parameter is fixed — either it is in the interval or it is not; the 95% describes the procedure's long-run coverage, not this realized interval
DThe statement would be correct only if the sample size were large enough for the CLT to apply
This is the single most common misinterpretation of confidence intervals. After the data are observed, L = 2.3 and U = 4.7 are fixed numbers, and θ is a fixed (unknown) number. There is no randomness left to assign a probability to. The 95% confidence level describes the *procedure*: if repeated many times, 95% of the resulting intervals would contain the true θ. No probability statement applies to any single computed interval.
Question 2 Multiple Choice
The test inversion approach to constructing a confidence interval produces a set of parameter values θ₀ that would not be rejected by a level-α test. How does this relate to the pivotal quantity approach?
AThey are unrelated — one is frequentist and one is Bayesian
BTest inversion applies only to one-sided tests; pivotal quantities produce two-sided intervals
CThey are mathematically equivalent and produce the same intervals
DTest inversion gives exact intervals while pivotal quantities only give approximate ones
The two constructions are formally equivalent. A pivotal quantity Q(X, θ) defines an interval by inverting P(a ≤ Q ≤ b) = 1-α into a range of θ values. The test inversion approach defines the confidence set as exactly those θ₀ for which a level-α test at the observed data would not reject — this is the same algebraic inversion. The equivalence reveals the deep connection between hypothesis testing and interval estimation.
Question 3 True / False
A 95% frequentist confidence interval and a 95% Bayesian credible interval answer fundamentally the same question about the parameter.
TTrue
FFalse
Answer: False
They answer different questions. The credible interval gives P(θ ∈ interval | data) = 0.95, treating θ as a random variable with a prior distribution — it makes a posterior probability statement about the parameter given the observed data. The confidence interval makes no probability statement about θ at all; it says that the *procedure* covers the true (fixed) θ in 95% of repetitions. The intervals may look numerically similar, especially with diffuse priors and large samples, but the philosophical interpretations are entirely distinct.
Question 4 True / False
The confidence level 1-α of a confidence interval refers to how often the interval construction procedure captures the true parameter in repeated sampling, not to the probability that a specific realized interval contains the parameter.
TTrue
FFalse
Answer: True
This is the correct frequentist interpretation. The randomness in a confidence interval comes from the data X: before observing data, [L(X), U(X)] is a random interval that covers θ with probability 1-α. After observing data, the interval is fixed. The 1-α coverage is a property of the procedure (the estimator, the pivotal quantity, the sample size), not of any individual result.
Question 5 Short Answer
Explain why it is incorrect to say 'there is a 95% probability that the parameter θ lies in this computed confidence interval,' once the data have been observed.
Think about your answer, then reveal below.
Model answer: After observing data, the confidence interval [L, U] is a pair of fixed numbers and θ is a fixed unknown constant. There is no random experiment being described — the probability is either 0 or 1 (θ either is or is not in the interval). The 95% refers to the long-run frequency: if the same procedure were repeated across many independent datasets, 95% of the resulting intervals would contain θ. To assign a probability to a specific realized interval requires treating θ as a random variable, which is the Bayesian approach (credible intervals), not the frequentist one.
Frequentist probability is defined over repeated sampling, not over unknown constants. The interval is what varies across repetitions; the parameter is fixed. This distinction also explains why two different 95% CIs from the same data (constructed by different methods) can have the same nominal coverage but very different actual properties — the coverage guarantee is about the procedure, not the particular numbers obtained.