How do I choose the right hypothesis test for FRM exam questions? I keep picking the wrong test statistic.

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AcadiFi · Accepted Answer

Choosing the correct hypothesis test is a high-frequency FRM Part I topic. The decision depends on three factors: what parameter you are testing, whether the population variance is known, and your sample size. Here is a systematic decision tree. ```mermaid graph TD A[What are you testing?] -->|Single mean| B{Population variance known?} A -->|Difference of two means| C{Variances known and equal?} A -->|Single variance| D[Chi-Square Test] A -->|Equality of two variances| E[F-Test] B -->|Yes| F[Z-Test] B -->|No| G{Sample size n >= 30?} G -->|Yes - use CLT| H[Z-Test approximately valid] G -->|No| I[T-Test with n-1 df] C -->|Yes, both known| J[Z-Test for difference] C -->|No, estimated from samples| K[T-Test - pooled or Welch] style A fill:#c9a84c,color:#0a0a0f style F fill:#1a3c2a,color:#fff style H fill:#1a3c2a,color:#fff style I fill:#1e3a5f,color:#fff style J fill:#1a3c2a,color:#fff style K fill:#1e3a5f,color:#fff style D fill:#2d1b4e,color:#fff style E fill:#2d1b4e,color:#fff ``` **Decision Rules Explained:** **1. Testing a Single Mean** - Population variance known: Use the z-statistic. z = (X-bar - mu0) / (sigma / sqrt(n)) - Variance unknown, n >= 30: z-test is approximately valid due to CLT, but many textbooks still prefer t-test - Variance unknown, n < 30: Must use t-test with (n-1) degrees of freedom **2. Testing Difference of Two Means** - Both variances known: z-test for the difference - Variances unknown but assumed equal: pooled t-test - Variances unknown and unequal: Welch's t-test (adjusted degrees of freedom) **3. Testing a Single Variance** - Always chi-square: chi-sq = (n-1) * s-squared / sigma0-squared with (n-1) df **4. Testing Equality of Two Variances** - Always F-test: F = s1-squared / s2-squared with (n1-1, n2-1) df **Worked Example:** Griffin Asset Management claims its equity fund generates an average monthly return of 1.2%. You collect 24 monthly returns and find a sample mean of 0.85% with a sample standard deviation of 2.10%. Test the claim at the 5% significance level. **Step 1:** Identify the test. Single mean, variance unknown, n = 24 < 30. Use t-test with 23 df. **Step 2:** Calculate the test statistic. t = (0.85 - 1.20) / (2.10 / sqrt(24)) = -0.35 / 0.4287 = **-0.816** **Step 3:** Find the critical value. Two-tailed test at 5%, t(0.025, 23) = +/- 2.069. **Step 4:** Decision. |-0.816| < 2.069, so we **fail to reject** the null hypothesis. There is insufficient evidence at the 5% level to conclude the fund's mean return differs from 1.2%. **Common Exam Traps:** 1. Using a z-test when n < 30 and variance is unknown — always use t-test 2. Forgetting that chi-square and F-tests are one-tailed by nature (variances are always positive) 3. Mixing up one-tailed and two-tailed critical values — read the question carefully 4. Not adjusting degrees of freedom for Welch's t-test when variances are unequal Practice more hypothesis testing problems in our FRM Part I Quantitative Analysis question bank.

How do I choose the right hypothesis test for FRM exam questions? I keep picking the wrong test statistic.

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