How to Choose the Right Hypothesis Test Formula on CFA Questions
Hypothesis testing does not start with memorizing a formula. It starts with reading the claim, identifying the baseline assumption, and deciding which decision route the exam is asking you to use. The formula is only the last step after you know what is being tested.
For CFA Level I, many hypothesis-testing questions are designed to see whether you understand the decision structure. Some questions require a test statistic. Others hand you a p-value, a critical value, or enough wording to decide whether a result is statistically significant. Treating every question as a formula hunt is slower and often less accurate.
The Three Decisions Before The Formula
Decision 1: What Is The Baseline Claim?
The null hypothesis is the baseline claim. It usually states equality, no change, no difference, or no effect. The alternative hypothesis is what the analyst needs evidence to support.
Original example:
Ridgeway Advisors claims that its new screening rule increases average monthly alpha for a small-cap strategy. The historical average alpha is 0.20 percent. A sample of portfolios using the new rule has an average alpha of 0.34 percent.
The null should not be "the rule works." The null is the baseline:
- Null hypothesis: average alpha is less than or equal to 0.20 percent.
- Alternative hypothesis: average alpha is greater than 0.20 percent.
That setup makes the test one-tailed because the analyst is asking whether performance improved, not whether it merely changed in either direction.
Decision 2: What Quantity Is Being Tested?
Once the null is clear, identify the parameter:
- One mean: average return, average duration error, average tracking difference.
- Difference in means: two manager groups, two periods, or two sample portfolios.
- Variance: dispersion or volatility claims.
- Correlation: whether a relationship differs from zero.
The parameter tells you the family of test. It also prevents the common error of using a single-mean formula for a comparison question.
Decision 3: What Evidence Did The Prompt Give You?
A prompt may give you any one of these:
- A sample mean and standard error, which points toward computing a test statistic.
- A test statistic and a critical value, which points toward a rejection-region comparison.
- A p-value and significance level, which points toward a p-value comparison.
- Pure wording about significance, which may require no calculation at all.
The Formula Map Candidates Actually Need
Single Mean: Known Versus Estimated Standard Deviation
If the question tests one population mean, the statistic is usually:
test statistic = (sample mean - hypothesized mean) / standard error
The key exam distinction is how the standard error is built:
- If population standard deviation is known, use a z-test framework.
- If population standard deviation is unknown and the sample standard deviation is used, use a t-test framework.
In most realistic CFA-style prompts, the population standard deviation is not known. If the prompt gives a sample standard deviation and sample size, expect a t-test unless the question explicitly tells you otherwise.
Original example:
An analyst tests whether average bid-ask spread for a trading strategy is below 18 basis points. A sample of 24 trades has an average spread of 15.9 basis points and a sample standard deviation of 5.4 basis points.
The denominator is 5.4 / square root of 24, and the reference distribution is t with 23 degrees of freedom. The important choice is not the arithmetic. It is recognizing that the sample standard deviation creates a t-test.
P-Value Route And Critical-Value Route Reach The Same Decision
The p-value route asks: if the null were true, how unusual is this sample result?
The critical-value route asks: is the test statistic far enough into the rejection region?
They are two ways to reach the same yes-or-no decision when the same significance level and tail direction are used.
| Evidence Provided | Decision Rule | Common CFA Mistake |
|---|---|---|
| p-value and alpha | Reject when p-value <= alpha | Thinking a larger p-value is stronger evidence |
| test statistic and critical value | Reject when statistic falls in rejection region | Ignoring one-tailed versus two-tailed direction |
| confidence interval | Reject when hypothesized value is outside interval | Using a one-sided claim with a two-sided shortcut blindly |
Degrees Of Freedom Are Not Decoration
Degrees of freedom determine which t critical value applies. For a one-sample mean test with sample standard deviation, degrees of freedom are usually n - 1.
If Ridgeway uses 24 observations, the reference is df = 23. A candidate who uses a z critical value because it is easier may get the wrong rejection decision when the sample is small.
A Worked Example
Fox Harbor Capital claims that its trade-cost model reduces average implementation shortfall below 30 basis points. A review of 36 trades shows an average shortfall of 27.4 basis points and a sample standard deviation of 9.0 basis points. Test at the 5 percent significance level. The one-tailed critical t-value for 35 degrees of freedom is approximately -1.69.
Step 1: Set Up The Hypotheses
- Null hypothesis: average shortfall is greater than or equal to 30 basis points.
- Alternative hypothesis: average shortfall is less than 30 basis points.
This is a left-tailed test because the claim is a reduction.
Step 2: Compute The Standard Error
standard error = 9.0 / square root of 36 = 1.5 basis points
Step 3: Compute The Test Statistic
t = (27.4 - 30.0) / 1.5 = -1.73
Step 4: Make The Decision
The statistic of -1.73 is less than the critical value of -1.69, so it falls in the rejection region. At the 5 percent level, the analyst rejects the null and concludes that the sample supports lower average implementation shortfall.
Step 5: Keep The Exam Framing Clean
The conclusion is statistical, not a guarantee that every future trade will be cheaper. A strong CFA answer states the decision, ties it to the evidence, and avoids overstating the business claim.
Exam Framing: What The Question Is Really Testing
CFA hypothesis-testing questions often test classification before computation:
- Can you identify the null as the baseline claim?
- Can you spot whether the alternative is one-tailed or two-tailed?
- Can you choose t versus z based on the data provided?
- Can you interpret a p-value without recomputing the statistic?
- Can you separate statistical significance from economic importance?
When a question gives a p-value of 0.08 at a 5 percent significance level, you do not need to reconstruct the entire test. The p-value already tells you not to reject at 5 percent. The exam is checking whether you trust the decision rule.
Fast Triage Checklist
Use This Order Under Time Pressure
- Translate the claim into null and alternative hypotheses.
- Identify the tested parameter.
- Determine tail direction.
- Read what evidence is already provided.
- Choose the shortest valid decision route.
- State the conclusion in terms of the null.
What To Avoid
Do not start by scanning memory for every formula in the reading. That turns a decision problem into a guessing game. The formula you need is usually obvious after the claim, parameter, tail, and evidence are clear.