What's the correct way to interpret a confidence interval? I keep losing marks on practice exams.

You're not alone — this is the single most commonly misinterpreted concept in CFA Level I Quantitative Methods.

The WRONG Interpretation (What You Wrote) 'There is a 95% probability that the population mean lies between 4.1 and 5.3.'

This is wrong because the population mean μ is a fixed (unknown) number, not a random variable. It either is or isn't in the interval — there's no probability about it.

The CORRECT Interpretation 'We are 95% confident that the interval [4.1, 5.3] contains the population mean.' This means: if we repeated this sampling procedure many times, approximately 95% of the resulting intervals would contain the true μ.

Building the Interval The formula for a confidence interval around a sample mean is:

CI = X̄ ± z_(α/2) × (σ / √n)

Example: Broadleaf Capital measures the average daily return on its mid-cap equity fund over 64 trading days. They find X̄ = 0.042% with σ = 0.16%. Construct a 99% confidence interval.

• z_(α/2) for 99% = 2.576
• Standard error = 0.16% / √64 = 0.02%
• CI = 0.042% ± 2.576 × 0.02% = 0.042% ± 0.05152%
• CI = [−0.00952%, 0.09352%]

Since the interval includes zero, we cannot conclude the fund has a statistically significant positive return at 99% confidence.

What Makes Intervals Wider or Narrower?

• Higher confidence level → wider interval (more certainty requires a bigger net)
• Larger sample size → narrower interval (more data = more precision)
• Larger σ → wider interval (more variability = less precision)

Exam Pitfall: If the question asks 'which of the following changes would most likely narrow the confidence interval?' the answer is almost always 'increase the sample size,' because it appears in the denominator under a square root, giving diminishing but reliable improvement.

Practice more questions like this in our CFA Level I question bank.