CFA Level 1 hypothesis testing and regression basics
Hypothesis testing evaluates evidence against a stated null hypothesis at a chosen significance level. Regression estimates a relationship between variables; a statistically significant slope does not, by itself, establish causation or a useful forecast.
The key ideas, at a glance
State the hypotheses
Read the visual, then follow the worked example below. Use the stated case assumptions.
Illustrative worked case
%
- p-value
- 3
- Significance level
- 5
p = 0.03 is below α = 0.05, so reject the stated null under the test assumptions. The p-value is not the probability that the null is true.
| Concept | Rule or decision | Common mistake |
|---|---|---|
| State the hypotheses | Choose a null and a one-sided or two-sided alternative before looking at the result. | Changing the alternative after seeing the data invalidates the intended test. |
| Choose the test | Match the statistic and distribution to the question, sample and assumptions. | A mean test and a regression slope test answer different questions. |
| Make the decision | Reject the null when the p-value is below the prespecified significance level. | Failure to reject is not proof that the null is true. |
| Interpret regression | The slope describes the estimated change in Y for a one-unit change in X. | Keep significance, explanatory power and causality separate. |
01State the hypotheses
Choose a null and a one-sided or two-sided alternative before looking at the result.
Watch for: Changing the alternative after seeing the data invalidates the intended test.
02Choose the test
Match the statistic and distribution to the question, sample and assumptions.
Watch for: A mean test and a regression slope test answer different questions.
03Make the decision
Reject the null when the p-value is below the prespecified significance level.
Watch for: Failure to reject is not proof that the null is true.
04Interpret regression
The slope describes the estimated change in Y for a one-unit change in X.
Watch for: Keep significance, explanatory power and causality separate.
Selected 2026 study concepts. Match your preparation to your registered exam-year curriculum. These notes are not complete curriculum coverage.
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Put the method to work
One worked example, from facts to answer
Given
A two-sided slope test has p = 0.03. The significance level chosen before testing is 0.05.
Method
0.03 < 0.05, so reject the null that the slope is zero under the test’s assumptions.
Answer and interpretation
The evidence supports a nonzero relationship. It does not prove that X causes Y or that the model will forecast well.
Original independent teaching example. Assumptions are stated above; this is not an official exam question or scoring key.
Apply the framework, step by step
Turn a hypothesis test or simple-regression output into a correctly qualified conclusion. Use Quantitative Methods for the broader return, probability and valuation foundations.
Before you start: Sample estimates, standard error, probability, significance levels and the difference between a dependent and independent variable.
METHOD 01
State the hypotheses
Choose a null and a one-sided or two-sided alternative before looking at the result.
State the null hypothesis and its alternative before using the results. A two-sided slope test asks whether the slope differs from a specified value; a one-sided test asks about a specified direction. The significance level sets the tolerated Type I error rate under the testing procedure. Do not select the direction or threshold after seeing which choice produces significance.
Avoid this mistake: Changing the alternative after seeing the data invalidates the intended test.
METHOD 02
Choose the test
Match the statistic and distribution to the question, sample and assumptions.
The test statistic measures how far the estimate is from the null value relative to its uncertainty. For a simple slope test, this often takes the form (estimated slope - hypothesized slope) / standard error. Match the reference distribution and degrees of freedom to the model and assumptions. A small sample or violated regression assumption can make a familiar shortcut inappropriate.
Avoid this mistake: A mean test and a regression slope test answer different questions.
METHOD 03
Make the decision
Reject the null when the p-value is below the prespecified significance level.
A p-value is calculated under the null model; it is not the probability that the null hypothesis is true. Compare it with the chosen significance level and report reject or fail to reject. Failing to reject can arise from weak evidence or low power. Statistical significance also differs from economic importance: a precisely estimated but tiny relationship may have little practical value.
Avoid this mistake: Failure to reject is not proof that the null is true.
METHOD 04
Interpret regression
The slope describes the estimated change in Y for a one-unit change in X.
In a simple regression, the slope is the estimated change in Y associated with a one-unit increase in X. The intercept is the fitted value when X is zero, which may lie outside the meaningful range. Inspect fit, residual behavior and the relevant assumptions. Correlation or a significant slope alone does not establish causation, and an in-sample fit does not guarantee out-of-sample forecasting performance.
Avoid this mistake: Keep significance, explanatory power and causality separate.
Apply it to a different case
- New facts
- Estimated slope is 0.80, hypothesized slope is 0, and its standard error is 0.20. The case gives a two-sided critical absolute statistic of 2.10.
- Working
- Test statistic = (0.80 - 0) / 0.20 = 4.00. Its absolute value exceeds 2.10.
- Conclusion
- Reject the zero-slope null under the stated test. The fitted relationship is positive; causal or forecasting claims require additional evidence.
Original teaching case under the stated assumptions, rather than an official exam question.
Questions about this method
- Does p = 0.03 mean the null has a 3% chance of being true?
- No. The p-value describes how unusual the observed test result, or a more extreme one, is under the null model and its assumptions. It is not a posterior probability of the null.
- Does failing to reject prove there is no relationship?
- No. It means the test did not provide enough evidence to reject the stated null at the selected level. Sample size, noise and model assumptions affect the result.
- What does a regression slope tell me?
- It estimates the change in the dependent variable associated with a one-unit change in the independent variable under the fitted model. It does not alone identify a causal effect.
Sources and curriculum scope
Official outlines establish curriculum scope. The explanations and worked examples are independent teaching material; the sources do not verify or endorse them.
- 2026 CFA Level 1 curriculum outline
- Official Level 1 format and curriculum
- 2027 CFA Level 1 curriculum outline
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