📖 假设检验导论：原假设与备择假设

Quantitative Methods — Hypothesis Testing · Lesson 1

1. From "Estimation" to "Decision": Why Hypothesis Testing?

In the previous six lessons (L125–L130), we learned how to: - Estimate the population mean μ using the sample mean x̄ (point estimation) - Construct a confidence interval to capture μ's plausible range (interval estimation)

But in the real world, we often need to make decisions, not just "estimate a range":

🔹 "Does this fund's annual return truly exceed the benchmark of 8%?" 🔹 "Is the new drug significantly more effective than the placebo?" 🔹 "Is there a fundamental difference between the returns of two investment strategies?"

These questions require not "what range is μ likely in?" but "can we conclude that μ > 8%?"

This is where hypothesis testing comes in.

2. The Core Idea of Hypothesis Testing

2.1 A Real-Life Analogy 💡

Imagine you are a judge:

🎯 Key Principle: Hypothesis testing uses the logic of "proof by contradiction" — we first assume H₀ is true, then check whether the sample data provides sufficiently strong contradictory evidence to overturn it.

2.2 The Three Core Elements

3. The Null Hypothesis H₀

3.1 Definition

The null hypothesis H₀ is the statement assumed to be true; we reject it only if the sample provides "sufficiently strong evidence to the contrary."

3.2 Why Is It Called "Null"?

"Null" means "no effect, no difference, no change" — it represents the status quo, the conservative stance.

3.3 H₀ Always Contains an Equality

This is a hard rule at CFA Level I:

✅ H₀: μ = 10     ✅ H₀: μ ≤ 10     ✅ H₀: μ ≥ 10
❌ H₀: μ ≠ 10     ❌ H₀: μ < 10      ❌ H₀: μ > 10

🔑 Reason: Testing requires computing probabilities (p-values) "under the assumption that H₀ is true." The equality gives H₀ a precise, computable benchmark.

4. The Alternative Hypothesis Hₐ

4.1 Definition

The alternative hypothesis Hₐ is the conclusion we lean toward when sample evidence is strong enough. It represents the claim the researcher genuinely wants to prove.

4.2 Hₐ Never Contains an Equality

✅ Hₐ: μ ≠ 10     ✅ Hₐ: μ > 10      ✅ Hₐ: μ < 10
❌ Hₐ: μ = 10     ❌ Hₐ: μ ≥ 10      ❌ Hₐ: μ ≤ 10

4.3 The Three Forms of Hₐ Determine the Test Direction

📌 The choice between two-tailed and one-tailed is NOT arbitrary — it is dictated by the research question, never chosen after seeing the data! (This is a key CFA exam point.)

5. How to Set Up Hypotheses: A Three-Step Practical Method

Step ①: Identify the "Claim" You Want to Prove

Write this claim in a form that does NOT contain an equality → this becomes your Hₐ candidate.

Step ②: H₀ Is the "Complement" of Hₐ

Replace Hₐ's ≠ / > / < with the complementary = / ≤ / ≥, ensuring H₀ contains an equality.

Step ③: Verify: H₀ contains equality; Hₐ does not

Example 1: Fund Manager Performance

Scenario: A fund manager claims their annual return exceeds the industry average of 8%. You want to test this claim.

⚠️ Note: If the question were simply "test whether it equals 8%," then Hₐ: μ ≠ 8%, H₀: μ = 8%.

Example 2: Quality Control

Scenario: A factory claims its products weigh 500g on average. The quality inspection department wants to test this claim.

Example 3: Process Improvement (Left-Tailed)

Scenario: A new process claims to reduce processing time, targeting an average below 30 minutes.

Example 4: Finance in Practice — Sharpe Ratio Test

Scenario: A strategy's backtested Sharpe ratio is 0.85. The classical view: if SR ≤ 0, the strategy has no excess return. You want to test whether the strategy has a positive Sharpe ratio.

6. Why Can't We Ever "Accept H₀"?

This is a classic CFA exam trap.

6.1 Only Two Possible Conclusions

6.2 Why Not Say "Accept H₀"?

Because "failing to reject" ≠ "confirming."

• Small sample size → may fail to reject H₀ even if it is false (lack of power)
• Failing to reject H₀ only means "there is currently not enough evidence to overturn it" — it does NOT mean "H₀ is true"

❌ Wrong: "We accept H₀, proving the mean equals 10"
✅ Correct: "At the 5% significance level, we fail to reject H₀"
✅ Plain English: "The data did not provide sufficient evidence to overturn H₀"

6.3 CFA Exam Takeaway

🔥 CFA standard phrasing is always "reject H₀" or "fail to reject H₀" — never "accept H₀."

7. From Question to Hypothesis: Common Exam Items and Pitfalls

7.1 Type 1: Write Hypotheses for a Given Scenario

Example: An investment bank analyst believes the P/E ratio of the tech sector differs from the market-wide average of 18x. State the null and alternative hypotheses.

Solution: - H₀: μ = 18 (sector P/E is the same as the market average) - Hₐ: μ ≠ 18 (sector P/E is different from the market average) - Test type: Two-tailed test (because "differs" carries no directional implication)

7.2 Type 2: Determine Whether to Use a One-Tailed Test

7.3 Type 3: Distinguishing the Roles of H₀ and Hₐ

💡 Mnemonic: What you want to prove goes in Hₐ; what needs to be refuted goes in H₀

• If you are a drug developer and want to prove "the new drug works":
• H₀: Drug is ineffective (μ = 0 or μ ≤ 0)
• Hₐ: Drug is effective (μ > 0)
• If you are a regulator and want to ensure "the new drug has no side effects":
• H₀: Drug is safe (side effects ≤ threshold)
• Hₐ: Drug is unsafe (side effects > threshold)

🎯 H₀ and Hₐ depend on your standpoint and the conclusion you are trying to prove!

8. The Hypothesis Testing Logical Flow (Preview)

1. State hypotheses
   H₀: μ = μ₀
   Hₐ: μ ≠ μ₀ (or > / <)

2. Choose significance level α (typically 0.05 or 0.01)
   ↓
3. Collect sample data → compute test statistic
   e.g., z = (x̄ − μ₀) / (σ/√n)
   ↓
4. Compute p-value or compare to critical value
   − p < α → reject H₀ ✅
   − p ≥ α → fail to reject H₀
   ↓
5. Draw conclusions (economic/financial interpretation)

📌 This lesson focuses on Step 1 (setting up hypotheses). Test statistics, p-values, and α levels will be covered in detail in L132.

9. Practice Questions

📝 Part A: Basic Concepts

Q1. Which of the following is true about the null hypothesis H₀?

A. H₀ is the claim the researcher wants to prove B. H₀ always contains "≠" C. H₀ is the statement assumed true unless sufficient evidence contradicts it D. H₀ and Hₐ can both be true simultaneously

Q2. In hypothesis testing, "rejecting H₀" means:

A. Proving that H₀ is false B. The sample data provide sufficiently strong evidence against H₀ C. H₀ is definitely false D. Hₐ is definitely true

Q3. Which of the following pairs of hypotheses is correctly specified?

A. H₀: μ = 5; Hₐ: μ = 6 B. H₀: μ ≠ 5; Hₐ: μ = 5 C. H₀: μ ≤ 5; Hₐ: μ > 5 D. H₀: μ < 5; Hₐ: μ > 5

📝 Part B: Hypothesis Setup

Q4. A researcher wants to test whether a new trading strategy generates positive excess returns (alpha). The correct hypotheses are:

A. H₀: alpha ≤ 0; Hₐ: alpha > 0 B. H₀: alpha = 0; Hₐ: alpha ≠ 0 C. H₀: alpha ≥ 0; Hₐ: alpha < 0 D. H₀: alpha < 0; Hₐ: alpha ≥ 0

Q5. A financial analyst wants to test whether the average annual return of the S&P 500 index differs from the historical average of 10%. The correct hypotheses are:

A. H₀: μ > 10%; Hₐ: μ ≤ 10% B. H₀: μ = 10%; Hₐ: μ ≠ 10% C. H₀: μ < 10%; Hₐ: μ ≥ 10% D. H₀: μ ≤ 10%; Hₐ: μ > 10%

Q6. A bank wants to verify whether a new approval system has reduced the average processing time to below 3 days. The correct hypotheses are:

A. H₀: μ ≥ 3; Hₐ: μ < 3 B. H₀: μ = 3; Hₐ: μ ≠ 3 C. H₀: μ ≤ 3; Hₐ: μ > 3 D. H₀: μ > 3; Hₐ: μ < 3

📝 Part C: Comprehensive Judgment

Q7. According to CFA Institute guidance, when conducting hypothesis tests, an analyst should state conclusions as:

A. "Accept H₀" or "reject H₀" B. "Accept Hₐ" or "reject Hₐ" C. "Reject H₀" or "fail to reject H₀" D. "Prove H₀" or "disprove H₀"

Q8. Which of the following situations is most appropriate for a two-tailed test?

A. Testing whether a new drug is better than an old one B. Testing whether a product's weight is below the standard C. Testing whether an index's return differs from the benchmark D. Testing whether employees' average overtime exceeds 10 hours

Q9. Regarding the relationship between H₀ and Hₐ, which of the following is INCORRECT?

A. H₀ and Hₐ are mutually exclusive B. H₀ and Hₐ together cover all possible values C. H₀ always represents the hypothesis the researcher wants to overturn D. In practice, we can simultaneously accept both H₀ and Hₐ

Q10. A portfolio manager claims their portfolio's Sharpe ratio is greater than 1.0. To test this claim, H₀ should be set as:

A. H₀: SR = 1.0 B. H₀: SR ≤ 1.0 C. H₀: SR ≥ 1.0 D. H₀: SR > 1.0

10. Answers and Explanations

Part A: Basic Concepts

Q1. Answer: C

H₀ is the "baseline statement assumed true" and is rejected only when sample evidence is strong enough. A describes Hₐ; B is wrong (H₀ contains equality, Hₐ does not); D is wrong (H₀ and Hₐ are mutually exclusive).

Q2. Answer: B

Rejecting H₀ means the contradiction between sample data and H₀ is too large to be attributed to random variation alone. This does not equal "proof" — statistical inference can never claim 100% certainty.

Q3. Answer: C

H₀ contains equality (≤), Hₐ does not (>). A is wrong because Hₐ includes =; B is wrong because H₀ includes ≠; D is wrong because neither H₀ nor Hₐ contains equality and they fail to cover all possible values.

Part B: Hypothesis Setup

Q4. Answer: A

Goal: prove "positive excess returns" → Hₐ: alpha > 0. H₀ takes the complement with equality → H₀: alpha ≤ 0.

Q5. Answer: B

"Differs from" → two-tailed test. H₀: μ = 10%; Hₐ: μ ≠ 10%. No directional implication → cannot use one-tailed.

Q6. Answer: A

Goal: prove "reduced to below 3 days" → direction is "less than" → Hₐ: μ < 3. Therefore H₀: μ ≥ 3.

Part C: Comprehensive Judgment

Q7. Answer: C

CFA standard: always say "reject H₀" or "fail to reject H₀," never "accept H₀."

Q8. Answer: C

"Differs from the benchmark" → a difference without direction → two-tailed. A and D are directional (better / exceeds), B is also directional.

Q9. Answer: D

H₀ and Hₐ cannot both be accepted simultaneously — they are mutually exclusive and exhaustive. D's error lies in "simultaneously accept," which is statistically impossible.

Q10. Answer: B

Manager claims SR > 1.0 → what you want to overturn is "SR ≤ 1.0" (the default position). So H₀: SR ≤ 1.0. This is the classic "verify exceptional performance" scenario: set the opposite of the claim as H₀.

11. Key Takeaways

12. Next Lesson Preview: L132

In the next lesson, we dive into p-values, significance level α, and Type I / Type II errors — answering the core question: "How strong does sample evidence need to be to qualify as 'strong enough'?"

📊 CFA Study Load for This Lesson: ~25 minutes reading + 10 practice questions

🏆 Congratulations! You've completed L131. See you tomorrow for L132!

CFA Level I · Quantitative Methods · Hypothesis Testing · Lesson 1 Generated for Ivan on 2026-08-06