📖 均值假设检验（z-test, t-test）

Quantitative Methods — Hypothesis Testing · Lesson 5

I. Recap: The Testing Framework Is Built

Today's core question: How exactly do we test whether a mean equals (is greater/less than) some value?

II. Two Tools for Testing Means

2.1 The Intuition

To test whether "the average annual return of Hong Kong funds is 5%":

1. Draw a sample, compute the sample mean x̄
2. Assess how far x̄ is from 5%
3. Determine if the gap is "random noise" or "real difference"

This judgment requires a test statistic. For means, we have two choices: z and t.

2.2 When to Use Which?

                Population σ known?
                  ├─ Yes → Z-Test
                  └─ No  → T-Test (use sample s instead of σ)

📌 CFA Level I golden rule: σ known → Z; σ unknown → T. Most financial scenarios have unknown σ → T-test dominates.

III. Z-Test

3.1 When Applicable

• Population normally distributed, or large sample (n ≥ 30, CLT)
• Population standard deviation σ is known
• Independent random sampling

3.2 Test Statistic

$$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$$

Where: - x̄ = sample mean - μ₀ = hypothesized population mean under H₀ - σ = known population standard deviation - n = sample size - SE = σ/√n (standard error)

3.3 Decision Rules at a Glance

3.4 Example: Flour Weight Inspection

A flour factory packages at 1000g standard, σ = 15g (known from experience). QC randomly samples 50 bags: x̄ = 995g. α = 0.05. Test for underweight.

H₀: μ ≥ 1000 (meets standard)
Hₐ: μ < 1000 (underweight) → left-tailed test

n = 50, σ = 15, x̄ = 995

z = (995 − 1000) / (15/√50)
  = −5 / 2.121
  = −2.357

α = 0.05, left-tailed Zα = −1.645
−2.357 < −1.645 → Reject H₀ ✅

Conclusion: Significant evidence the machine fills below 1000g. Calibration recommended.

🎯 Z-test feature: if σ is known, the z statistic follows an exact normal distribution with fixed critical values.

IV. T-Test

4.1 Why Do We Need the T-Test?

In reality, σ is almost never known.

When we substitute s (sample standard deviation) for σ: - The numerator x̄ − μ₀ is still approximately normal - But the denominator gains an extra layer of uncertainty (s itself varies) - → Result: the test statistic follows a t-distribution, not a normal distribution

Z statistic:      (x̄ − μ₀) / (σ/√n)  → Normal distribution
T statistic:      (x̄ − μ₀) / (s/√n)  → t-distribution
                           ↑
                    This s introduces extra uncertainty

4.2 Characteristics of the t-Distribution

                t-Distribution vs Normal

    Normal (higher peak, thinner tails)
       ╱          ╲          t(df=2) — fattest tails
      ╱            ╲         t(df=5) — moderate
     ╱              ╲        t(df=30) — approaching normal
────┴────────────────┴────

📌 At the same α, t critical value > Z critical value. The extra uncertainty from using s in place of σ requires a larger "safety margin."

4.3 Test Statistic

$$t_{n-1} = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$

Where: - s = sample standard deviation = $\sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$ - df = n − 1

4.4 Decision Rules

p < α → Reject H₀ (same as Z-test)

V. Case Study: Fund Alpha Test

5.1 Data

A fund has 36 months of monthly excess returns averaging 0.65%, with s = 2.80%. H₀: μ = 0 (no alpha), Hₐ: μ ≠ 0 (two-tailed), α = 0.05

n = 36, x̄ = 0.65%, s = 2.80%

SE = 2.80% / √36 = 0.4667%

t = (0.65% − 0) / 0.4667% = 1.393

df = 35
t₀.₀₂₅,₃₅ ≈ 2.030 (table or calculator)

|t| = 1.393 < 2.030 → Fail to reject H₀ ❌

Conclusion: Monthly excess of 0.65%, but volatility is too high — statistically insufficient to prove alpha exists.

5.2 Intuitive Explanation

0.65% monthly average — looks decent on the surface
But 2.80% std dev over 36 months — extreme volatility
→ This 0.65% could just be a lucky 36-month run
→ Need a longer track record or larger alpha to prove it

VI. Z vs T Quick Comparison

VII. Key Critical Value Comparison (α = 0.05)

📌 Smaller n (smaller df) → larger t critical value → higher bar to reject H₀ → "sample size penalty"

VIII. The Triple Effect of Sample Size

🎯 Increasing sample size is the "universal remedy" for boosting test power — simultaneously reduces SE, lowers critical values, and increases power.

IX. Classic CFA Traps

Trap 1: When to Use Z-Test?

❌ Sample size > 30 → automatically use Z ✅ Only use Z when σ is known! n > 30 but σ unknown → still use T

The Z vs T criterion is whether σ is known, NOT how large n is.

Trap 2: Using s with Z Critical Values

❌ If the question gives s instead of σ, but you apply Z critical values anyway ✅ s → must use T, look up critical values from the t-table (by n−1)

Trap 3: Wrong Degrees of Freedom

❌ df = n (always treating n as df) ✅ One-sample mean test: df = n − 1

Trap 4: Thinking n > 30 Means t = Normal

❌ n = 35 → mentally default to Z critical values ✅ At n = 35, t₀.₀₂₅,₃₄ ≈ 2.032 > 1.96 — still a difference! Only at n ≈ 120+ do they become virtually identical.

Trap 5: p-Value Determination

❌ Compute the t statistic, then look it up in the Z-table ✅ t statistics must be looked up in the t-table (by df), otherwise p-values are incorrect

X. Complete Hypothesis Testing Template

Standard five-step procedure for a mean hypothesis test:

Step 1: State the hypotheses
  H₀: μ = μ₀ (or ≤, ≥)
  Hₐ: μ ≠ μ₀ (or >, <)

Step 2: Select the test statistic
  ├─ σ known → z = (x̄ − μ₀) / (σ/√n)
  └─ σ unknown → t = (x̄ − μ₀) / (s/√n), df = n−1

Step 3: Determine significance level and critical value
  α = 0.05 → look up Z-table or t-table

Step 4: Compute and compare
  Calculate statistic → compare with critical value / find p-value

Step 5: Make decision and interpret
  "At α = 0.05, [reject / fail to reject] H₀.
   There [is / is not] sufficient evidence that..."

📝 Practice Questions

Part A: Choose the Test Method

Q1. Given σ = 8, n = 25, test μ = 100. Use:

A. Z-test B. T-test C. Chi-square test D. F-test

Q2. σ unknown, n = 50, x̄ = 3.2, s = 1.5. Test μ = 3. Use:

A. Z-test, use normal table B. T-test, df = 49 C. T-test, df = 50 D. Z-test, because n > 30

Part B: Hypothesis Setup & Decision

Q3. An analyst wants to test whether fund alpha is positive. H₀ should be:

A. H₀: μ = 0 B. H₀: μ ≤ 0 C. H₀: μ ≥ 0 D. H₀: μ ≠ 0

Q4. n = 16, x̄ = 21.3, s = 3.8, H₀: μ = 20, Hₐ: μ ≠ 20, α = 0.05. The t-value is approximately:

A. 0.34 B. 1.37 C. 2.06 D. 2.74

Part C: Comprehensive Judgment

Q5. n = 25, x̄ = 48, s = 10, H₀: μ = 50, Hₐ: μ < 50, α = 0.05. Which is correct?

A. Z-test, z = −1.0 B. T-test, df = 24 C. T-test, df = 25 D. Z-test, because σ is unknown

Q6. Regarding the t-distribution, which is correct?

A. The t-distribution always has a higher peak than the normal distribution B. As df increases, the t-distribution approaches the normal distribution C. The t-test cannot be used with small samples D. When n > 30, the t critical value equals the normal critical value

Q7. For the same dataset, which factors will increase the t statistic?

I. Increase n II. Increase s III. Increase |x̄ − μ₀| IV. Increase α

A. I and III B. II and IV C. I, II, III D. All of the above

Q8. σ known vs σ unknown — with the same data, which makes it easier to reject H₀?

A. σ known (Z-test) B. σ unknown (T-test) C. Depends on n D. Exactly the same

📊 Answers & Explanations

📌 Key Memory Card

🔑 σ known = Z; σ unknown = T. This decision rule is the CFA Level I calculation question's mandatory first step.

🧠 Extended Thinking: Why T-Tests Dominate in Real Finance?

σ (population standard deviation) in finance:
  - True std dev of stock returns? Unknown — we can only infer it
  - Industry-average fund alpha volatility? No official data
  - Mortgage rate dispersion? History changes, the population value doesn't exist

→ You are always using s to estimate σ
→ So the T-test is the absolute workhorse of CFA Level I calculation questions
→ The Z-test primarily appears in conceptual questions: "If σ = 10..."

💡 Understanding the Z vs T difference is fundamentally about understanding the extra uncertainty that comes from "known vs estimated."

Next up: L136 — Tests for Mean Differences (Paired and Independent Samples), learning to compare means of two populations.