How do I interpret z-scores in hypothesis testing — and when should I use a z-test vs. a t-test?

Great question — the z-score vs. t-test distinction trips up a lot of CFA candidates. Here's the clear breakdown:

What a Z-Score Represents A z-score measures how many standard deviations an observation (or sample mean) falls from the population mean. The formula is:

z = (X̄ − μ₀) / (σ / √n)

Where X̄ is the sample mean, μ₀ is the hypothesized population mean, σ is the known population standard deviation, and n is the sample size.

When to Use Z vs. T

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Worked Example: Nelton Industries claims its quarterly earnings growth averages 4.2%. You sample 36 quarters and find X̄ = 3.6% with a known σ = 2.4%.

• H₀: μ = 4.2%
• Hₐ: μ ≠ 4.2% (two-tailed)
• z = (3.6 − 4.2) / (2.4 / √36) = −0.6 / 0.4 = −1.50

At a 5% significance level (two-tailed), the critical values are ±1.96. Since |−1.50| < 1.96, we fail to reject H₀. The sample doesn't provide enough evidence that the true mean differs from 4.2%.

Key Exam Tips:

1. A z-score of ±1.96 corresponds to a 95% confidence level (two-tailed) — memorize this.
2. At ±2.58 you're at 99% confidence.
3. With large samples (n ≥ 30), the t-distribution converges toward the z-distribution, so the choice matters less.
4. The CFA exam will always specify whether σ is known — read the vignette carefully.

For more practice with hypothesis testing, explore our CFA Level I Quantitative Methods course.