Correct Interpretation of a Frequentist Confidence Interval
What is the correct frequentist interpretation of a $100(1-\alpha)\%$ confidence interval for a parameter $\theta$?
Specifically: is it correct to say that $\theta$ lies in the interval with probability $1-\alpha$? Why or why not? What does the interval actually tell you?
Hints
- Ask yourself: in the frequentist framework, is the parameter $\theta$ random or fixed? This determines where the probability statement can come from.
- The randomness in a confidence interval comes from the sample, not the parameter. The interval $[L(X), U(X)]$ is a random quantity because $X$ is random; $\theta$ is a constant.
- The correct statement is about the procedure: $P(L(X) \leq \theta \leq U(X)) = 1-\alpha$ before data collection. After you compute a specific interval, there is no meaningful probability left -- $\theta$ either is or is not inside it.
Worked Solution
How to Think About It: This is one of the most commonly misunderstood concepts in statistics. The tempting (wrong) interpretation is: 'There is a $95\%$ probability that $\theta$ is inside this interval.' But from the frequentist perspective, $\theta$ is a fixed, non-random constant -- it either is or is not in any given interval, with probability 0 or 1. The probability $1-\alpha$ refers to the procedure, not to any particular realized interval.
Key Insight: In frequentist statistics, randomness comes from the data (the sample), not from the parameter. The confidence interval is random (because it is computed from random data); the parameter is not. The correct interpretation must live in the world of hypothetical repeated sampling.
The Correct Interpretation:
A $100(1-\alpha)\%$ confidence interval $[L(X), U(X)]$ satisfies:
$$P\left(L(X) \leq \theta \leq U(X)\right) = 1 - \alpha$$
where the probability is over the randomness in the data $X$. That is:
> If you repeated the experiment many times -- drawing a new sample each time, computing a new interval each time -- then approximately $100(1-\alpha)\%$ of those intervals would contain the true $\theta$.
Once you have computed a specific interval from your data, say $[2.1, 4.7]$, you cannot say 'there is a 95% probability that $\theta$ is between 2.1 and 4.7.' At that point, $\theta$ either is or is not in $[2.1, 4.7]$ -- no probability is involved.
What is and is not random: - The interval endpoints $L(X)$ and $U(X)$ are random (they depend on the sample $X$). - The parameter $\theta$ is fixed and non-random (frequentist assumption). - The statement '$\theta$ falls in the interval' is what has a $1-\alpha$ probability -- before the data is collected.
Contrast with Bayesian credible intervals:
A Bayesian $95\%$ credible interval does allow the statement 'there is a 95% probability that $\theta$ is in this interval' -- because in the Bayesian framework, $\theta$ itself has a probability distribution (the posterior). Frequentist and Bayesian intervals are computed differently and have different interpretations.
Answer: The correct interpretation is: the procedure that produced this interval will cover the true $\theta$ in $100(1-\alpha)\%$ of repeated experiments. You cannot assign a probability to whether $\theta$ is inside any specific computed interval.
Intuition
The confidence interval misinterpretation is so pervasive that even published papers get it wrong. The conceptual trap is conflating 'the probability that the interval contains $\theta$' (a statement about the random interval, before data collection -- this is $1-\alpha$) with 'the probability that $\theta$ is in this particular interval' (a statement about the fixed parameter given a realized interval -- this is either 0 or 1 in the frequentist framework).
If you want to make probability statements about unknown parameters, you need the Bayesian framework, where $\theta$ is treated as a random variable with a prior distribution. A $95\%$ Bayesian credible interval genuinely means 'given the data, I believe $\theta$ is in this range with 95% posterior probability.' The frequentist confidence interval is a statement about long-run coverage of a procedure -- a subtly but importantly different claim. In quantitative finance, this distinction matters most when interpreting model parameters and their uncertainty -- especially when the parameter space is small and the idea of 'repeated experiments' is not meaningful.