V1 Module 7 Errata: Estimation and Hypothesis Testing

1. Strong Skewness Is Treated as a CLT Violation

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

"Significant skewness or extreme values, such as those seen with gold prices, violate these assumptions, making the resulting confidence intervals less reliable."

Strong skewness or extreme observations can make a finite-sample normal approximation poor and require a larger sample. They do not, by themselves, violate the CLT when observations are independent and identically distributed and have finite variance.

Correct reading: Strong skewness or extreme observations can require a larger sample before the normal approximation is reliable; they are not automatically violations of the CLT assumptions.

Candidates should distinguish a slow finite-sample approximation from a failure of the theorem's conditions.

2. VaR Is Misclassified as a Confidence Interval

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

3. Probability Sampling Is Said to Give Everyone an Equal Chance

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

4. The Simple-Random-Sampling Paragraph Calls Selection Bias Sampling Error

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

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

Sampling error is random sample-to-sample variation caused by observing only a subset of the population. Selection bias is a systematic distortion caused by the sampling mechanism. An unbiased random sample can therefore have sampling error without being biased.

Correct reading: An unbiased random sample can still have sampling error because only a subset of the population is observed. Selection bias is a separate systematic problem in how observations enter the sample.

Candidates should distinguish unavoidable random sampling variation from a selection process that is systematically displaced.

5. Systematic Sampling Omits Its Random Start

Curriculum location: The Central Limit Theorem, Confidence Intervals, and Sampling, systematic-sampling discussion, p. 322.

"Systematic sampling selects every $k$th member of a sample population."

"Since every member has an equal chance of being selected, the sample is likely to be more representative of the population than a simple random sample."

Selecting every $k$th unit is not a probability design until the starting position is randomized. With a fixed start, one subset is selected with certainty and the others cannot be selected. Periodicity aligned with $k$ can also make the sample less representative, so systematic sampling is not generally superior to simple random sampling.

Correct reading: Randomly choose the start among the first $k$ positions, then take every $k$th member and check for relevant periodicity.

Candidates should include the random start and check periodicity when identifying or implementing a valid systematic sample.

6. Right-Tailed Tests Use the Wrong Decision Rule

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

"If the absolute value of the test statistic exceeds the critical value, we reject the null hypothesis in favor of the alternative hypothesis, considering the result statistically significant."

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

"At $\alpha =0.05$, reject $H_0$ if $t<-1.6628$."

"At $\alpha =0.01$, reject $H_0$ if $t<-2.3705$."

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

• At $\alpha =0.05$, reject $H_0$ if $t>1.6628$.
• At $\alpha =0.01$, reject $H_0$ if $t>2.3705$.

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.

7. Population Sizes Are Used Instead of Sample Sizes

Curriculum location: Hypothesis Testing and Parametric Testing, difference-in-means degrees of freedom, p. 344.

"The degrees of freedom for the test are determined by the two population sizes and are equal to $n_1+n_2-2$."

The formula uses $n_1$ and $n_2$, which the same page defines as the two sample sizes. The sizes of the underlying populations do not determine the pooled two-sample t-test degrees of freedom.

Correct reading: The degrees of freedom are determined by the two sample sizes and equal $n_1+n_2-2$.

Candidates should use the observations in the two samples—not the sizes of the populations from which those samples were drawn—to determine the degrees of freedom.

8. The General p-Value Definition Is Written as a Right-Tail Rule

Curriculum location: Hypothesis Testing and Parametric Testing, p-value definition, p. 354.

"It is the probability of observing a test statistic at least as high as the sample’s assuming that $H_0$ is true."

“At least as high” describes only a right-tailed test. The same paragraph also gives left-tailed and two-tailed formulas, for which the relevant outcomes are respectively low enough or extreme in either direction.

Correct reading: A p-value is the probability, assuming $H_0$ is true, of observing a test statistic at least as extreme as the one observed, with extremeness determined by the alternative hypothesis and the test direction.

Candidates should identify the alternative hypothesis before deciding which tail or tails count as evidence against $H_0$.

9. T.DIST.RT(prob, df) Reverses the Function's Input and Output

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

Description cell:

"Returns the right tail probability of a t-distribution with a given input, the inverse of the Student’s t-distribution for a one-tailed test"

Excel syntax cell:

"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.

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

Curriculum location: Non-Parametric Tests, Spearman calculation procedure, 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, p. 363.

"The null hypothesis, $H_0:r_S=0$."

"The alternative hypothesis, $H_A:r_S\ne 0$."

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:

$$H_0:{\rho }_S=0,H_A:{\rho }_S\ne 0.$$

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

Scope: Module 7, pp. 309–379 (printed page numbers).

What this review covers. Errors in the printed curriculum that would change a candidate's answer or understanding: wrong numbers, wrong formulas, reversed logic, and statements that contradict the module's own data. It does not list spelling mistakes, equation-numbering slips, or wording that is loose but defensible.

References

• CFA curriculum refresher reading: Estimation and Inference
• CFA curriculum refresher reading: Hypothesis Testing
• NIST/SEMATECH: Confidence Intervals
• NIST/SEMATECH: Two-Sample t-Test
• NIST/SEMATECH: Critical Values and p-Values
• NIST/SEMATECH: Rank Correlation Independence Test
• Statistics Canada: Probability Sampling
• Statistics Canada: Non-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 CFA Institute errata notice, and inclusion here must not be read as CFA Institute confirmation, endorsement, or approval.