定量方法（Quantitative Methods）— 假设检验 · 第 4 课
一、上节课回顾：你知道检验会出错
L133 教会了我们 Type I（α，冤枉好人）和 Type II（β，放过坏人）两种错误：
- Type I = α：H₀ 真但拒绝了 — 假阳性
- Type II = β：H₀ 假但保留了 — 假阴性
- Power = 1 − β：正确拒绝假 H₀ 的能力
- α 和 β 此消彼长，唯有增大 n 才能同时优化
二、为什么检验要分「单尾」和「双尾」？
2.1 核心直觉
假设检验的问题本质上是「我要怀疑的方向是什么？」
回顾 L131 的框架：Hₐ（备择假设）是一个有方向的主张。 这个主张的方向，决定了拒绝域的形状。
2.2 三种提问方式决定两种尾巴
| 提问方式 | 对应的 Hₐ | 检验类型 |
|---|---|---|
| 「这个策略有没有 Alpha？」 | μ ≠ 0（不管是正还是负） | 🔄 双尾检验 |
| 「这个策略超额收益是否大于 0？」 | μ > 0 | ➡️ 右尾检验（单尾） |
| 「这只基金亏损是否低于同业？」 | μ < benchmark_loss | ⬅️ 左尾检验（单尾） |
🎯 Hₐ 的方向决定了尾巴的数量和位置。有方向的 Hₐ → 单尾，无方向的 Hₐ → 双尾。
三、双尾检验（Two-Tailed Test）
3.1 结构
H₀: μ = μ₀
Hₐ: μ ≠ μ₀
3.2 拒绝域
双尾检验把 α 一分为二，左右各 α/2。α = 0.05 → 临界值 = ±Z_{α/2} = ±1.96
正态分布 N(0,1)
╱ ╲
╱ ╲
┌────────╱ ╲────────┐
α/2=2.5% α/2=2.5%
────┴──────────────────┴──────────────────┴────
-1.96 0 +1.96
3.3 决策规则
- 如果 |检验统计量| > |临界值| → 拒绝 H₀
- 如果 p 值 < α → 拒绝 H₀
- 📌 双尾 p = P(|Z| > |z₀|) = 2 × P(Z > |z₀|)
3.4 金融场景
场景 1：两只基金表现是否不同？ H₀: μ_A - μ_B = 0, Hₐ: μ_A - μ_B ≠ 0 → 双尾。只想知道「有没有差异」。
场景 2：检验股票收益率是否等于零 H₀: μ_returns = 0, Hₐ: μ_returns ≠ 0 → 双尾。涨跌都有意义。
四、单尾检验（One-Tailed Test）
4.1 右尾检验（Upper-Tail Test）
H₀: μ ≤ μ₀
Hₐ: μ > μ₀
正态分布 N(0,1)
╱ ╲
╱ ╲────────┐
╱ ╲ α=5%
────────────┴──────────────────┴────
0 +1.645
α = 0.05 → 临界值 = 1.645（全部 α 在右侧）
金融场景： 基金经理声称策略有正 Alpha → 右尾。只关心「超过 0」。
4.2 左尾检验（Lower-Tail Test）
H₀: μ ≥ μ₀
Hₐ: μ < μ₀
正态分布 N(0,1)
╱ ╲
┌────────╱ ╲
α=5%
────┴──────────────────┴─────────────
-1.645 0
α = 0.05 → 临界值 = -1.645（全部 α 在左侧）
金融场景： 监管机构证明产品 VaR 低于上限 → 左尾。只关心「低于」。
五、关键数值速查表（CFA 必背）
正态分布 Z 临界值
| α | 单尾 Zα | 双尾 Zα/₂ |
|---|---|---|
| 0.10 | ±1.282 | ±1.645 |
| 0.05 | ±1.645 | ±1.96 |
| 0.01 | ±2.326 | ±2.576 |
📌 同 α 下，单尾更容易拒绝 H₀（门槛 1.645 < 1.96）。⚠️ 但必须 Ha 真有方向才能用单尾！
六、单尾 vs 双尾的 p 值差异
关键公式
双尾 p = 2 × P(Z > |z₀|) = 单尾 p × 2
右尾 p = P(Z > z₀)
左尾 p = P(Z < z₀)
🔥 核心案例
z₀ = 1.80
单尾（右尾）p ≈ 0.0359 → α=0.05 → 拒绝 H₀ ✅
双尾 p = 2 × 0.0359 ≈ 0.0718 → α=0.05 → 不拒绝 H₀ ❌
🚨 同一个 z₀，单尾能做到显著，双尾做不到！这就是「双尾检验惩罚」。
七、什么时候用单尾/双尾？
决策流程
1. 问题有没有明确方向？
├─ 有（「更好」「大于」「小于」）→ 第 2 步
└─ 无（「不同」「有差异」）→ 双尾
2. 方向？
├─ > → 右尾
└─ < → 左尾
金融场景速判
| 场景 | Hₐ | 类型 |
|---|---|---|
| 策略优于基准 | μ > μ_bm | 右尾 |
| 错误率低于阈值 | μ < 5% | 左尾 |
| A/B 测试广告 | μ_A ≠ μ_B | 双尾 |
| 股价偏离 0 | μ ≠ 0 | 双尾 |
| 风险超标检测 | μ > limit | 右尾 |
常见误区
| 误区 | 真相 |
|---|---|
| ❌ 看数据后再决定单尾/双尾 | 🚨 Data snooping！学术不端！必须事前确定 |
| ❌ 单尾比双尾好（更容易显著） | ⚠️ 方向错 → 永远发现不了真相 |
| ❌ 左尾右尾只是符号不同 | 🔑 H₀/Hₐ 完全不同，拒绝域完全相反 |
| ❌ 双尾更严格所以更好 | ⚠️ 有方向时浪费 Power，效率更低 |
八、实战案例：同一数据，不同结论
某基金 60 个月，月均收益 0.80%，标准差 3.00%。基准月收益 = 0%。
n = 60, x̄ = 0.80%, s = 3.00%
SE = 3.00% / √60 = 0.3873%
t = (0.80% - 0) / 0.3873% = 2.066
右尾检验（H₀: μ ≤ 0, Hₐ: μ > 0） 临界值 t₀.₀₅,₅₉ ≈ 1.671 → 2.066 > 1.671 → 拒绝 H₀ ✅
双尾检验（H₀: μ = 0, Hₐ: μ ≠ 0） 临界值 t₀.₀₂₅,₅₉ ≈ ±2.001 → |2.066| > 2.001 → 拒绝 H₀ ✅
如果 t = 1.90 呢？
| 检验 | t = 2.066 | t = 1.90 |
|---|---|---|
| 右尾 (临界 1.671) | ✅ 拒绝 | ✅ 拒绝 |
| 双尾 (临界 ±2.001) | ✅ 拒绝 | ❌ 不拒绝 |
📌 边界情况，结论完全不同！
九、单尾陷阱：方向错误
真实效果为负（μ < 0），但用右尾 Hₐ: μ > 0
t = -2.5 → 负值，无法落入右侧拒绝域
→ 永远不拒绝 H₀，错过负向信号！
改用双尾：|t| = 2.5 > 1.96 → 拒绝 H₀！发现了显著负效果。
🚨 单尾选错方向 = 永远看不到反方向的真相。 💡 不确定方向 → 用双尾。事后换方向 → 必须重新收集数据。
十、CFA 考点总结
| 层次 | 考点 |
|---|---|
| 识别 | Hₐ 含 ≠ → 双尾；Hₐ 含 > → 右尾；Hₐ 含 < → 左尾 |
| 临界值 | α=0.05：单尾 1.645 / 双尾 1.96（死记！） |
| p 值 | 双尾 p = 单尾 p × 2 |
| 结论 | 单尾更易拒绝 H₀；方向错则永远发现不了反方向效果 |
| 原则 | 必须事前确定，不能看数据后选 |
| 情境 | 超额收益 → 右尾；合规 → 左尾；两策略比较 → 双尾 |
📝 课堂练习
Part A：识别检验类型
Q1. H₀: μ = 10, Hₐ: μ ≠ 10。这是：
A. 右尾检验 B. 左尾检验 C. 双尾检验 D. 无法判断
Q2. 分析师想证明基金 Sharpe Ratio 高于行业平均 0.5。合适的检验是：
A. H₀: μ ≤ 0.5, Hₐ: μ > 0.5（右尾） B. H₀: μ ≥ 0.5, Hₐ: μ < 0.5（左尾） C. H₀: μ = 0.5, Hₐ: μ ≠ 0.5（双尾） D. 以上都不对
Q3. 检验 H₀: μ ≥ 100, Hₐ: μ < 100。这是：
A. 右尾检验 B. 左尾检验 C. 双尾检验 D. 无法判断
Part B：临界值与 p 值
Q4. α = 0.05 下，双尾检验 Z 临界值约为：
A. 1.282 B. 1.645 C. 1.96 D. 2.576
Q5. 某检验右尾 p 值为 0.03，α = 0.05。对应双尾 p 值为：
A. 0.015 B. 0.03 C. 0.06 D. 0.12
Q6. 检验统计量 z = 2.10，α = 0.05。以下正确的是：
A. 右尾检验拒绝 H₀，双尾检验也拒绝 H₀ B. 右尾检验拒绝 H₀，双尾检验不拒绝 H₀ C. 右尾检验不拒绝 H₀，双尾检验拒绝 H₀ D. 两种检验都不拒绝 H₀
Part C：综合应用
Q7. 某检验报告双尾 p 值 = 0.04，α = 0.05。结论是：
A. 拒绝 H₀，因为 0.04 < 0.05 B. 不拒绝 H₀，因为 0.04/2 = 0.02 < 0.05 C. 拒绝 H₀，因为 0.04/2 = 0.02 < 0.05 D. 无法判断
Q8. 关于单尾和双尾，正确的是：
A. 单尾检验总比双尾好，因为更容易显著 B. 看到数据后如果 p 接近 0.05，可以临时换单尾 C. 如果 Hₐ 有明确方向，应使用对应单尾检验 D. 左尾和右尾本质相同，只是临界值符号不同
📊 答案与解析
| 题号 | 答案 | 解析 |
|---|---|---|
| Q1 | C | Hₐ 含 ≠，标准双尾结构。拒绝域在分布两侧。 |
| Q2 | A | "高于" → 有明确方向 → 右尾。H₀ 是现状（≤ 0.5），Hₐ 是研究主张（> 0.5）。 |
| Q3 | B | H₀: μ ≥ 100 是"不低于"（现状）；Hₐ: μ < 100 是"低于"（主张），方向向左 → 左尾。 |
| Q4 | C | α = 0.05 双尾 → ±1.96。CFA 一级必须死记！ |
| Q5 | C | 双尾 p = 单尾 p × 2。0.03 × 2 = 0.06 > 0.05 → 双尾下不显著。 |
| Q6 | A | z = 2.10。右尾临界 1.645（2.10 > 1.645 ✅）；双尾临界 1.96（2.10 > 1.96 ✅）。两种都拒绝。 |
| Q7 | A | ⚠️ 关键陷阱！双尾 p = 0.04 已经是双侧之和，直接和 α 比较即可！不需要再 ×2！ |
| Q8 | C | A 错：单尾不一定好。B 错：事后选尾巴是 data snooping。D 错：左尾右尾 H₀/Hₐ 完全不同。C 正确。 |
📌 本课核心记忆卡
| 概念 | 一句话 |
|---|---|
| 双尾检验 | Hₐ 含 ≠，拒绝域在两侧，更保守 |
| 单尾检验 | Hₐ 含 > 或 <，拒绝域在一侧，同 α 下更容易拒绝 |
| 临界值 α=0.05 | 单尾 1.645 / 双尾 1.96 |
| p 值关系 | 双尾 p = 单尾 p × 2 |
| 选择原则 | 事前根据 Hₐ 方向决定，不能看数据后改 |
| 方向错误 | 单尾选错 → 永远看不到反方向的真实效果 |
| CFA 陷阱 | 给的是双尾 p 就不需要再 ×2；给的是单尾 p 才需要考虑 ×2 |
🔑 记住：尾巴的数量和位置，由你的研究问题（Hₐ）决定，不由数据决定。
下节课预告：L135 — 均值检验（Z 检验与 t 检验），学习如何选择和使用正确的检验统计量。
Quantitative Methods — Hypothesis Testing · Lesson 4
I. Review: You Know Tests Make Errors
L133 taught us about Type I (α, convicting the innocent) and Type II (β, letting the guilty go):
- Type I = α: H₀ is true but rejected — false positive
- Type II = β: H₀ is false but retained — false negative
- Power = 1 − β: The ability to correctly reject a false H₀
- α and β trade off against each other; only increasing n improves both
II. Why Do We Distinguish "One-Tailed" from "Two-Tailed"?
2.1 Core Intuition
Hypothesis testing is fundamentally about: "Which direction am I questioning?"
Recall from L131: Hₐ (the alternative hypothesis) is a directional claim. The direction of this claim determines the shape of the rejection region.
2.2 Three Questions, Two Tail Types
| Question | Corresponding Hₐ | Test Type |
|---|---|---|
| "Does this strategy have Alpha?" | μ ≠ 0 (positive OR negative) | 🔄 Two-Tailed |
| "Does this strategy's excess return exceed 0?" | μ > 0 | ➡️ Right-Tailed (One-Tailed) |
| "Is this fund's loss lower than peers?" | μ < benchmark_loss | ⬅️ Left-Tailed (One-Tailed) |
🎯 Hₐ's direction determines the number and position of tails. Directional Hₐ → One-Tailed; Non-directional Hₐ → Two-Tailed.
III. Two-Tailed Test — "I only care about difference, not direction"
3.1 Structure
H₀: μ = μ₀
Hₐ: μ ≠ μ₀
3.2 Rejection Region
A two-tailed test splits α equally into both tails, α/2 on each side.
N(0,1)
╱ ╲
╱ ╲
┌────────╱ ╲────────┐
α/2=2.5% α/2=2.5%
───────┴──────────────┴──────────────┴───────
-1.96 0 +1.96
- α = 0.05 → 2.5% in each tail
- Critical value = ±Z_{α/2} = ±1.96
3.3 Decision Rule
- If |test statistic| > |critical value| → Reject H₀
- If p-value < α → Reject H₀
- 📌 Two-tailed p-value: p = P(|Z| > |z₀|) = 2 × P(Z > |z₀|)
3.4 Finance Examples
Example 1: Do two funds perform differently? H₀: μ_A − μ_B = 0, Hₐ: μ_A − μ_B ≠ 0 → Two-tailed: "is there a difference?" (either direction).
Example 2: Does a stock's return differ from zero? H₀: μ_returns = 0, Hₐ: μ_returns ≠ 0 → Two-tailed: both gains and losses matter.
IV. One-Tailed Test — "I Have a Clear Direction"
4.1 Right-Tailed (Upper-Tail) Test
H₀: μ ≤ μ₀
Hₐ: μ > μ₀
N(0,1)
╱ ╲
╱ ╲────────┐
╱ ╲ α=5%
────────────────┴──────────────┴────
0 +1.645
α = 0.05 → Critical value = 1.645 (all α on the right)
Finance: A fund manager claims positive Alpha → Right-tailed. We only care about "exceeding 0."
4.2 Left-Tailed (Lower-Tail) Test
H₀: μ ≥ μ₀
Hₐ: μ < μ₀
N(0,1)
╱ ╲
┌────────╱ ╲
α=5%
────────┴──────────────┴─────────────
-1.645 0
α = 0.05 → Critical value = −1.645 (all α on the left)
Finance: Regulator proves VaR is below the cap → Left-tailed. Only care about "below."
V. Critical Value Quick Reference (Must Memorize!)
Z Critical Values (Normal Distribution)
| α | One-Tailed Zα | Two-Tailed Zα/₂ |
|---|---|---|
| 0.10 | ±1.282 | ±1.645 |
| 0.05 | ±1.645 | ±1.96 |
| 0.01 | ±2.326 | ±2.576 |
📌 At the same α, a one-tailed test rejects H₀ more easily (threshold 1.645 < 1.96). ⚠️ But only use one-tailed when Hₐ truly has a clear direction!
VI. p-Value Differences: One-Tailed vs Two-Tailed
Key Formulas
Two-tailed p = 2 × P(Z > |z₀|) = One-tailed p × 2
Right-tailed p = P(Z > z₀)
Left-tailed p = P(Z < z₀)
🔥 Critical Example
z₀ = 1.80
One-tailed (right) p ≈ 0.0359 → α=0.05 → Reject H₀ ✅
Two-tailed p = 2 × 0.0359 ≈ 0.0718 → α=0.05 → Do NOT reject H₀ ❌
🚨 Same z₀ — one-tailed can reject H₀, two-tailed cannot! This is the "two-tailed penalty."
VII. When to Use One-Tailed vs Two-Tailed?
Decision Flow
1. Does my research question have a clear direction?
├─ Yes ("better", "greater than", "less than") → Step 2
└─ No ("different", "not equal to") → Two-tailed
2. What direction?
├─ Greater → Right-tailed
└─ Less → Left-tailed
Quick Finance Reference
| Scenario | Hₐ | Type |
|---|---|---|
| Strategy outperforms benchmark | μ > μ_bm | Right-tailed |
| Error rate below threshold | μ < 5% | Left-tailed |
| A/B testing two ads | μ_A ≠ μ_B | Two-tailed |
| Stock price deviates from 0 | μ ≠ 0 | Two-tailed |
| Risk exceeds limit | μ > limit | Right-tailed |
Common Misconceptions
| Misconception | Truth |
|---|---|
| ❌ Choose tail after seeing data | 🚨 Data snooping! Academic misconduct! Decide before data. |
| ❌ One-tailed is better (easier significance) | ⚠️ Wrong direction → you never discover the truth. |
| ❌ Left/right tail just differ by sign | 🔑 H₀/Hₐ are entirely different; rejection regions are opposite. |
| ❌ Two-tailed is stricter, so always better | ⚠️ With a directional expectation, it wastes Power. |
VIII. Case Study: Same Data, Different Conclusions
A fund over 60 months: average monthly return = 0.80%, σ = 3.00%. Benchmark return = 0%.
n = 60, x̄ = 0.80%, s = 3.00%
SE = 3.00% / √60 = 0.3873%
t = (0.80% − 0) / 0.3873% = 2.066
Right-tailed (H₀: μ ≤ 0, Hₐ: μ > 0) Critical t₀.₀₅,₅₉ ≈ 1.671 → 2.066 > 1.671 → Reject H₀ ✅
Two-tailed (H₀: μ = 0, Hₐ: μ ≠ 0) Critical t₀.₀₂₅,₅₉ ≈ ±2.001 → |2.066| > 2.001 → Reject H₀ ✅
What if t = 1.90?
| Test | t = 2.066 | t = 1.90 |
|---|---|---|
| Right-tailed (crit 1.671) | ✅ Reject | ✅ Reject |
| Two-tailed (crit ±2.001) | ✅ Reject | ❌ Do not reject |
📌 At the margin, conclusions diverge completely!
IX. The One-Tailed Trap: Wrong Direction
True effect is negative (μ < 0), but using right-tailed Hₐ: μ > 0
t = −2.5 → negative value, can never fall in right rejection region
→ Never reject H₀, miss the negative signal entirely!
Switch to two-tailed: |t| = 2.5 > 1.96 → Reject H₀! Discovered significant negative effect.
🚨 One-tailed + wrong direction = you never see the opposite truth. 💡 Unsure about direction → use two-tailed. Change direction later → recollect data.
X. CFA Exam Key Points
| Level | Key Point |
|---|---|
| Identification | Hₐ has ≠ → two-tailed; > → right-tailed; < → left-tailed |
| Critical Values | α=0.05: one-tailed 1.645 / two-tailed 1.96 (memorize!) |
| p-Value | Two-tailed p = one-tailed p × 2 |
| Conclusion | One-tailed rejects more easily; wrong direction misses opposite effects |
| Principle | Decide before seeing data — no post-hoc switching |
| Application | Excess return → right-tail; Compliance → left-tail; Strategy comparison → two-tail |
📝 Practice Questions
Part A: Identify the Test Type
Q1. H₀: μ = 10, Hₐ: μ ≠ 10. This is:
A. Right-tailed test B. Left-tailed test C. Two-tailed test D. Cannot determine
Q2. An analyst wants to prove a fund's Sharpe Ratio exceeds the industry average of 0.5. The appropriate test is:
A. H₀: μ ≤ 0.5, Hₐ: μ > 0.5 (right-tailed) B. H₀: μ ≥ 0.5, Hₐ: μ < 0.5 (left-tailed) C. H₀: μ = 0.5, Hₐ: μ ≠ 0.5 (two-tailed) D. None of the above
Q3. Test: H₀: μ ≥ 100, Hₐ: μ < 100. This is:
A. Right-tailed test B. Left-tailed test C. Two-tailed test D. Cannot determine
Part B: Critical Values & p-Values
Q4. At α = 0.05, the two-tailed Z critical value is approximately:
A. 1.282 B. 1.645 C. 1.96 D. 2.576
Q5. A test has a right-tailed p-value of 0.03, α = 0.05. The corresponding two-tailed p-value is:
A. 0.015 B. 0.03 C. 0.06 D. 0.12
Q6. Test statistic z = 2.10, α = 0.05. Which is correct?
A. Right-tailed rejects H₀, two-tailed also rejects H₀ B. Right-tailed rejects H₀, two-tailed does not reject H₀ C. Right-tailed does not reject H₀, two-tailed rejects H₀ D. Neither test rejects H₀
Part C: Comprehensive Application
Q7. A test reports a two-tailed p-value = 0.04, α = 0.05. The conclusion is:
A. Reject H₀, because 0.04 < 0.05 B. Do not reject H₀, because 0.04/2 = 0.02 < 0.05 C. Reject H₀, because 0.04/2 = 0.02 < 0.05 D. Cannot determine
Q8. Regarding one-tailed and two-tailed tests, which is correct?
A. One-tailed is always better because it's easier to achieve significance B. If p is close to 0.05 after seeing data, you can switch to one-tailed C. If Hₐ has a clear direction, the corresponding one-tailed test should be used D. Left-tailed and right-tailed are essentially the same, only the critical value sign differs
📊 Answers & Explanations
| # | Answer | Explanation |
|---|---|---|
| Q1 | C | Hₐ contains ≠ — classic two-tailed structure. Rejection region on both sides. |
| Q2 | A | "Exceeds" → clear direction → right-tailed. H₀ is status quo (≤ 0.5), Hₐ is research claim (> 0.5). |
| Q3 | B | H₀: μ ≥ 100 is "not less than" (status quo); Hₐ: μ < 100 is "less than" (claim), direction is left → left-tailed. |
| Q4 | C | α = 0.05 two-tailed → ±1.96. Must memorize for CFA Level I! |
| Q5 | C | Two-tailed p = one-tailed p × 2. 0.03 × 2 = 0.06 > 0.05 → not significant under two-tailed. |
| Q6 | A | z = 2.10. Right-tail crit 1.645 (2.10 > 1.645 ✅); Two-tail crit 1.96 (2.10 > 1.96 ✅). Both reject. |
| Q7 | A | ⚠️ Key trap! The reported p = 0.04 is already two-tailed (the sum of both sides). Compare directly to α. Do NOT multiply by 2 again! |
| Q8 | C | A is wrong: one-tailed is not always better. B is wrong: post-hoc switching is data snooping. D is wrong: left/right have completely different H₀/Hₐ. C is correct. |
📌 Key Takeaways
| Concept | In One Sentence |
|---|---|
| Two-Tailed | Hₐ contains ≠, rejection on both sides, more conservative |
| One-Tailed | Hₐ contains > or <, rejection on one side, easier to reject at same α |
| Critical values α=0.05 | One-tailed 1.645 / Two-tailed 1.96 |
| p-Value relationship | Two-tailed p = One-tailed p × 2 |
| Selection principle | Decide based on Hₐ direction BEFORE seeing data |
| Wrong direction | One-tailed + wrong direction = never see the opposite truth |
| CFA trap | If given two-tailed p, compare directly to α — do NOT multiply by 2 again! |
🔑 Remember: the number and position of tails are determined by your research question (Hₐ), not by the data.
Next: L135 — Tests of the Mean (Z-test and t-test), learning how to choose and use the correct test statistic.