CFA Level I 2027 Errata: Estimation and Hypothesis Testing

1. Probability Is Assigned to Fixed Parameters

Curriculum location: The Central Limit Theorem, Confidence Intervals, and Sampling, CLT applications, printed p. 315.

"This technique assesses the probability that a specific population parameter is in some hypothesized range."

Curriculum location: Non-Parametric Tests, final Module Summary, printed p. 372.

"Confidence intervals provide a range of values, based on the sample estimate, in which the underlying population parameters lie with a specified probability."

Both statements move probability from the sampling procedure or the data to a fixed but unknown population parameter. In the module's frequentist framework, a hypothesis test evaluates sample evidence against a null hypothesis, while a confidence level describes the long-run coverage of an interval-producing procedure. Neither is a posterior probability assigned to the parameter.

Correct reading: A hypothesis test uses sample evidence to assess whether to reject a null hypothesis at a specified significance level. A confidence interval gives a range of plausible values for an unknown population parameter at a stated confidence level. Across repeated samples, intervals constructed by a 95% procedure would contain the true parameter approximately 95% of the time.

For candidates, this distinction keeps confidence levels, p-values, and posterior probabilities from being treated as interchangeable.

2. The Sample Mean Is Confused with the Test Statistic

Curriculum location: Hypothesis Testing and Parametric Testing, test-statistic selection, printed p. 330.

"The sampling distribution of the mean return on an asset is t-distributed when the population standard deviation is unknown and normally distributed. It is Z-distributed when the population standard deviation is known."

For a mean test, the known-versus-unknown standard-deviation distinction determines the reference distribution of the standardized test statistic:

$$Z=\frac{\bar X-\mu_0}{\sigma/\sqrt n}, \qquad t=\frac{\bar X-\mu_0}{s/\sqrt n}.$$

It is not the sample mean itself that switches from a Z-distribution to a t-distribution.

Correct reading: In the curriculum's normally distributed population setting, use a Z-statistic when $\sigma$ is known and a t-statistic when $\sigma$ is estimated by $s$.

This distinction determines which test statistic and critical distribution a candidate should use.

3. VaR Is Misclassified as a Confidence Interval

Curriculum location: The Central Limit Theorem, Confidence Intervals, and Sampling, VaR discussion, printed p. 319.

"VaR represents the lower bound in a one-sided confidence interval around the expected value."

The passage classifies a loss quantile as an interval estimate. VaR identifies a one-sided loss threshold for a stated horizon and confidence level, whereas a confidence interval estimates an unknown parameter.

Correct reading: VaR is a one-sided loss threshold for a stated holding period and confidence level, not a confidence interval for a population parameter.

Candidates should distinguish a risk threshold from an interval that describes uncertainty about an estimated parameter.

4. Right-Tailed Tests Use the Wrong Decision Rule

Curriculum location: Hypothesis Testing and Parametric Testing, Establish the Decision Rule, printed p. 333.

"If the absolute value of the test statistic exceeds the critical value, we reject the null hypothesis."

Curriculum location: Hypothesis Testing and Parametric Testing, difference-in-means case, printed p. 345. The stated alternative is right-tailed.

$$t<-1.6628 \qquad\text{and}\qquad t<-2.3705.$$

Taking an absolute value removes direction. That shortcut is valid for a symmetric two-tailed test, but not for a one-tailed test. The negative cutoffs in the worked example likewise define a left-tailed rejection region even though the stated alternative requires a right-tailed region.

Correct reading: Reject the null hypothesis when the test statistic enters the rejection region implied by the alternative hypothesis. Use the absolute-value shortcut only for a symmetric two-tailed test. For this right-tailed example, the correct rejection rules are

$$t>1.6628 \qquad\text{and}\qquad t>2.3705.$$

Tail direction controls the sign of the cutoff, the rejection inequality, and the p-value calculation. Correct arithmetic cannot rescue a decision rule that points in the wrong direction.

5. T.DIST.RT Is Misidentified as an Inverse Function

Curriculum location: Hypothesis Testing and Parametric Testing, calculator-function table, printed p. 357.

"T.DIST.RT ... the inverse of the Student's t-distribution for a one-tailed test" with syntax T.DIST.RT(prob, df).

The calculator entry reverses input and output by describing a tail-probability function as an inverse function.

Correct reading: T.DIST.RT(x, df) takes a t value and degrees of freedom and returns the probability to the right. An inverse function instead takes a probability and returns a cutoff.

Candidates need this distinction to choose between calculating a tail probability and retrieving a critical value.

6. Probability Sampling Is Confused with Equal-Probability Sampling

Curriculum location: The Central Limit Theorem, Confidence Intervals, and Sampling, probability-sampling introduction, printed p. 322.

"Probability sampling ensures every member of the population has an equal chance of selection, leading to a representative sample."

Equal selection probability is a feature of simple random sampling, not a requirement for every probability-sampling design. Other probability samples may assign different selection probabilities across population members.

Correct reading: Probability sampling gives every member of the population a known, nonzero probability of selection. Those probabilities need not be equal.

This distinction prevents candidates from rejecting valid probability-sampling designs merely because their selection probabilities differ.

7. Sampling Error Is Confused with Selection Bias

Curriculum location: The Central Limit Theorem, Confidence Intervals, and Sampling, simple-random-sampling introduction, printed p. 322.

"This method can introduce sampling error because biased samples can generate biased estimates of the true parameter values."

Sampling error is random variation caused by observing only part of a population. Selection bias is systematic distortion caused by how observations enter the sample. A correctly executed random sample can still have sampling error.

Correct reading: Simple random sampling selects population members by chance with equal selection probabilities. Sampling error is the random difference between a sample statistic and its population parameter because only a subset of the population is observed.

For candidates, sampling error and selection bias call for different diagnoses and cannot be used as synonyms.

8. Non-Probability Sampling Is Treated as a CLT Failure

Curriculum location: The Central Limit Theorem, Confidence Intervals, and Sampling, non-probability-sampling introduction, printed p. 323.

"Consequently, the CLT does not reliably apply here."

The CLT concerns the sampling distribution of a statistic under mathematical conditions. Whether a sample supports inference to a target population is a separate sampling-design question.

Correct reading: Non-probability sampling may produce a sample that does not represent the target population, limiting population inference. This selection problem is distinct from the mathematical conditions of the CLT.

Candidates should identify whether a question concerns representativeness or a sampling-distribution approximation before selecting the relevant principle.

9. Skewness Is Confused with Nonstationarity

Curriculum location: The Central Limit Theorem, Confidence Intervals, and Sampling, gold-price case conclusion, printed p. 319.

"Significant skewness or extreme values ... violate these assumptions." The passage then refers to the problem as "non-stationarity."

Strong skewness can slow practical convergence toward a normal approximation. Nonstationarity instead means that the data-generating process changes over time.

Correct reading: Strong skewness or extreme observations can require a larger sample before the normal approximation is reliable. Nonstationarity is a separate issue in which the data-generating process changes over time.

Candidates should not treat distribution shape and stability over time as two names for the same problem.

10. Spearman's Shortcut Omits the No-Ties Condition

Curriculum location: Non-Parametric Tests, Spearman calculation procedure, printed p. 360. The original sequence discusses average ranks for ties and then gives the following shortcut without a no-ties condition.

$$r_S=1-\frac{6\sum d_i^2}{n(n^2-1)}.$$

The rank-difference shortcut is exact when there are no tied ranks. When ties occur, assigning average ranks changes the variance structure behind that shortcut.

Correct reading: When there are no tied ranks, the rank-difference shortcut may be used directly. When ties are present, assign average ranks and calculate Spearman correlation as the Pearson correlation of the assigned ranks.

Before applying the shortcut, candidates should check whether tied ranks are present.

11. Sample Statistics Are Used in Population Hypotheses

Curriculum location: Non-Parametric Tests, Spearman hypothesis test, printed p. 363.

$$H_0:r_S=0, \qquad H_A:r_S\ne0.$$

The hypotheses concern the population rank-correlation parameter, not the statistic observed in one sample. Using $r_S$ in the hypotheses confuses the quantity being tested with the evidence used to test it.

Correct reading: State the hypotheses using the population rank-correlation parameter:

$$H_0:\rho_S=0, \qquad H_A:\rho_S\ne0.$$

The sample statistic $r_S$ is then calculated from the observed data.

Candidates should identify whether each symbol denotes a population parameter or a sample statistic before writing hypotheses.

Complete Module 7 Body-Text Errata Index

Errata scope

• Curriculum: CFA Level I 2027 Curriculum
• Volume: Volume 1, Quantitative Methods
• Module: Module 7, Estimation and Hypothesis Testing
• Topics covered: The Central Limit Theorem, Confidence Intervals, and Sampling; Hypothesis Testing and Parametric Testing; Non-Parametric Tests
• Source reviewed: The 72-page Module 7 curriculum PDF

The table below lists all confirmed source-authored body-text errors identified across the three topics. Repeated instances of the same defect, such as the four VaR misclassifications, are consolidated into one row. Import-only defects and end-of-module question-set issues are excluded.

Page references use the printed curriculum page numbers.

References

• CFA curriculum refresher reading: Estimation and Inference
• CFA curriculum refresher reading: Hypothesis Testing
• NIST/SEMATECH: Frequentist and Bayesian Parameter Interpretation
• NIST/SEMATECH: Confidence Intervals
• NIST/SEMATECH: Rank Correlation Independence Test
• Statistics Canada: Probability Sampling
• Microsoft Support: T.DIST.RT Function
• Federal Reserve: VaR-Based Measure

This article is an independent candidate-focused analysis of confirmed errors in the CFA Level I 2027 Curriculum, Volume 1 Quantitative Methods, Module 7 Estimation and Hypothesis Testing. It is not an official curriculum errata notice.