Statistical Inference — The CFA Level 1 Remix

"You have data. You have dreams. Statistical inference tells you which dreams are plausible and which are delusions dressed in spreadsheets."

You already know the scaffolding from Quantitative Methods > Probability Concepts and Basic Statistics: random variables, sampling distributions, the central limit theorem, and summary statistics. Now we move from describing the world to making decisions about it. Statistical inference is where we use sample data to make probabilistic claims about populations — with humility, math, and a small dose of skepticism.

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Why this matters for CFA candidates (and ethical finance types)

• Investment decisions, risk management, and performance claims all rest on inference: is a strategy really beating a benchmark, or are we witnessing lucky noise?
• Ethics and Professional Standards matters here. Misusing inference — p-hacking, data dredging, cherry-picking windows — can mislead clients and violate the Code and Standards. Be rigorous, transparent, and honest about uncertainty.

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Big ideas, fast

• Point estimate: a single-number guess for a population parameter (eg, sample mean x̄ as an estimate of population mean μ).
• Interval estimate: a range that plausibly contains the parameter (eg, 95% confidence interval).
• Hypothesis test: structured decision rule to evaluate a specific claim about a population parameter.
• Type I error (α): rejecting a true null hypothesis. Think: false alarm.
• Type II error (β) and power (1 − β): failing to reject a false null; power is the test's ability to detect a true effect.

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Common tools and formulas

Confidence interval for a mean

• If population sd σ is known (rare in finance):

CI: x̄ ± z* (σ / √n)

• If σ unknown (usual case):

CI: x̄ ± t* (s / √n)

Where z* is the standard normal critical value and t* is from Student's t with n − 1 degrees of freedom.

Hypothesis testing for a mean

Null and alternative depending on claim. Example formats:

• Two-sided: H0: μ = μ0 vs Ha: μ ≠ μ0
• One-sided: H0: μ ≤ μ0 vs Ha: μ > μ0 (claim of outperformance)

Test statistic (σ unknown):

t = (x̄ − μ0) / (s / √n)

Compare t to t-critical or compute p-value.

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Quick decision flow (aka your cheat-code)

1. State H0 and Ha explicitly.
2. Choose α (commonly 0.05).
3. Compute test statistic.
4. Find p-value or critical region.
5. Make decision and interpret in context.

Given sample {data}, test H0: μ = μ0
compute x̄, s, n
t = (x̄ - μ0) / (s / sqrt(n))
if p-value < α then reject H0 else do not reject H0

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Z vs t (handy comparison)

Situation	Known σ?	Distribution	Use when...
z-test	Yes	Standard normal	Rare in finance; sometimes for large-sample approximations
t-test	No	Student's t (df = n − 1)	Usual: use when σ is unknown and n is moderate/small

Remember: as n grows, t approaches z.

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A financey example — does the fund manager actually beat the benchmark?

Scenario: Manager claims fund mean monthly return > 0.5% (benchmark). Sample: n = 50 months, sample mean x̄ = 0.7% (0.007), sample sd s = 4% (0.04). Test at α = 0.05.

1. H0: μ ≤ 0.5% (0.005). Ha: μ > 0.5% (one-sided).
2. Test statistic:

t = (0.007 − 0.005) / (0.04 / sqrt(50))
  = 0.002 / 0.005657 ≈ 0.354

3. t ≈ 0.354, df = 49. One-sided p-value ≈ 0.36. Not significant.

Interpretation: With this sample, there is weak evidence that the manager beats the benchmark. Don’t hype the result in a marketing pitch. Full-stop.

(If you see tiny p-values in fund marketing, ask about multiple testing and look for transparency — remember Ethics.)

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Errors, power, and sample size — the awkward trio

• Lower α reduces Type I errors but increases Type II errors unless you increase n.
• Power depends on effect size, n, α, and variability. More variability means you need a larger n to detect the same effect.
• In finance, small effect sizes are common; plan sample sizes accordingly and be suspicious of small-sample claims.

Practical tip: if the cost of missing a true effect is high, choose higher power (eg, 0.8) and compute required n.

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Assumptions checklist (do not skip)

• Random sampling or independent observations.
• For t-tests: approximate normality of sampling distribution. Thanks, CLT — n ≥ 30 often saves us, but check for heavy tails.
• Homoscedasticity for two-sample tests, unless using robust versions.

If assumptions fail, use nonparametric tests or bootstrap confidence intervals.

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Ethical spotlight

Bad inference plus marketing equals regulatory trouble and harmed clients.

• Always report the method, assumptions, sample size, and p-values, not just whether the result was 'significant'.
• Avoid slicing data until something looks significant (data snooping). Pre-specify tests when possible.
• Disclose multiple comparisons and adjust for them (eg, Bonferroni) or, better, avoid undisclosed hunting.

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Common misunderstandings (for extra credit)

• A 95% confidence interval does not mean 95% probability that the true parameter lies in that interval for the specific interval observed. It means the procedure will capture the true parameter in 95% of repeated samples.
• A p-value is not the probability that H0 is true. It is the probability of observing data as extreme as yours, assuming H0 is true.
• 'Not significant' ≠ 'no effect'. It might be 'insufficient evidence'. Consider effect size and CI width.

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Closing rally: what to memorize and what to apply

• Memorize the testing steps, CI formulas, and the distinction between z and t.
• Learn to compute and interpret p-values, Type I/II errors, and power intuitively.
• Always pair statistical conclusions with transparent ethical reporting. Your credibility and clients depend on it.

Final thought: In finance, claims without rigorous inference are like investment memos written in glitter and hope. Use inference to move from intuition to justified action — and keep your feet, and your clients, safe on the ground.