z-statistic
A process has known σ = 27. A sample of n = 36 gives x̄ = 186.6. Compute the z-statistic for H₀: μ = 183.
Answer and worked solution
Answer: 0.8
z = (x̄ − μ₀)/(σ/√n) = (186.6 − 183)/(27/6) = 0.8. |z| > 1.96 rejects at 5% (two-sided).
Estimation & testing: interview questions and key results
z-tests, CIs, p-values, sample size. The results to know for quant trading interviews, then a worked example of each of the 6 question types. Free, no sign-up.
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Key results
- SE of the mean: σ/√n. 95% CI: x̄ ± 1.96·SE.
- p-value: probability of data at least this extreme if H₀ is true. It is not P(H₀ true).
- Sharpe t-stat ≈ SRannual·√years.
Worked examples (6)
Try each one before opening the solution. In the app, every type generates fresh numbers each time.
Confidence-interval width
A sample of n = 16 has standard deviation 5. Using z = 1.96, what is the half-width of the 95% confidence interval for the mean?
Answer and worked solution
Answer: 2.45
Half-width = 1.96·s/√n = 1.96·5/4 = 2.45. Quadrupling n only halves the width.
Two-sided p-value
A test statistic is z = 1. What is the two-sided p-value?
Φ(0.5)=0.6915 · Φ(1)=0.8413 · Φ(1.5)=0.9332 · Φ(1.645)=0.95 · Φ(1.96)=0.975 · Φ(2)=0.9772 · Φ(2.5)=0.9938 · Φ(3)=0.9987
Answer and worked solution
Answer: 0.31731
p = 2·(1 − Φ(|z|)) = 2·(1 − 0.84134) ≈ 0.31731.
Sample size
Returns have σ = 40 bps. How many observations do you need so the 95% CI for the mean has half-width at most 3 bps? (z = 1.96)
Answer and worked solution
Answer: 683
Need 1.96σ/√n ≤ E ⇒ n ≥ (1.96·40/3)² = 682.95, so n = 683 (round up).
SE of a proportion
A signal predicted direction correctly in 10% of n = 100 trades. What is the standard error of that hit rate?
Answer and worked solution
Answer: 0.03
SE = √(p(1−p)/n) = √(0.1·0.9/100) ≈ 0.03.
Sharpe t-statistic
A backtest shows an annualised Sharpe ratio of 1 over 1 year. Approximately what is the t-statistic that the true mean return is non-zero?
Answer and worked solution
Answer: 1
t = mean/(σ/√N) = SRperiod·√N = SRannual·√years = 1·√1 ≈ 1. Short backtests rarely prove anything, and testing many strategies makes it worse.
Other topics
Dice & coins · Cards & combinatorics · Conditional & Bayes · Expected value & games · Random walks & martingales · Markov chains & order statistics · Options · Classic brainteasers · Number sequences · Distributions · Variance & covariance · Regression & correlation · Returns, risk & sizing · Time series
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