CFA Level I · CFA Level I Exam

Estimation and Hypothesis Testing: formula sheet

Full chapter guide

Key formulas

Systematic sampling interval: k = population size ÷ sample size; Choose a random start between 1 and k, then take every kth member.
Stratified sample size per stratum: Stratum sample = total sample × (stratum size ÷ population size); Proportional allocation keeps the sample mix equal to the population mix. Very small strata may round to zero, so round up or set a minimum allocation.
Sampling error: Sampling error = sample statistic − population parameter; Arises from chance even with a proper random sample. It is not the same as bias.
Bias to method matching: Failed entities missing → survivorship; future information used → look-ahead; repeated testing on same data → data snooping; unusual or short period → time-period; Learn the single clue word for each bias.
Standard error (population σ known): σx̄ = σ ÷ √n; σ is the population standard deviation, n is the sample size.
Standard error (σ unknown): sx̄ = s ÷ √n; s is the sample standard deviation. This is the usual case in practice.
CLT distribution of the sample mean: x̄ ~ approximately N(μ, σ² ÷ n); Needs random sampling, finite variance and a large n (rule of thumb n ≥ 30).
Standardized sample mean: z = (x̄ − μ) ÷ (σ ÷ √n); Use this to find probabilities for a sample mean. With s instead of σ, a t-statistic is used in testing.
Standard error of the sample mean (analytical): s_X̄ = s ÷ √n; The benchmark formula. Resampling is used when no such formula is practical.
Bootstrap standard error: SE(bootstrap) = standard deviation of the B resampled statistics; Each resample is drawn with replacement and has size n. The usual convention is the sample standard deviation of the B statistics.
Jackknife resample count and size: n resamples, each of size n − 1; Each resample omits exactly one observation. Result is deterministic.
Jackknife mean of the statistic: θ̄ = (1 ÷ n) × Σ θ_i; θ_i is the statistic computed with observation i left out.
Jackknife standard error: SE(jackknife) = √[ ((n − 1) ÷ n) × Σ(θ_i − θ̄)² ]; Sum runs over i = 1 to n. It is not a plain standard deviation of the θ_i: the sum of squares is scaled by (n − 1) ÷ n.
Test statistic: Test statistic = (sample statistic − hypothesized value) ÷ standard error of the sample statistic; For a mean with unknown population variance, use t = (x̄ − μ0) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Hypothesis forms: Two-tailed: H0: θ = θ0, Ha: θ ≠ θ0. One-tailed: H0: θ ≤ θ0, Ha: θ > θ0 (or H0: θ ≥ θ0, Ha: θ < θ0); H0 always holds the equality sign. Ha holds the direction.
Critical value decision rule: Reject H0 if the test statistic is beyond the critical value (in the rejection region); Two-tailed: use α ÷ 2 in each tail. One-tailed: use all of α in one tail.
p-value decision rule: Reject H0 if p-value < α; Otherwise fail to reject H0. Never say you accept H0.
Error probabilities and power: P(Type I error) = α; P(Type II error) = β; Power = 1 − β; Power is the probability of rejecting H0 when H0 is false.
Confidence interval link: Two-tailed test at α: reject H0 if the hypothesized value lies outside the (1 − α) confidence interval; Confidence level = 1 − α.
t-test for a single mean: t = (x̄ − μ0) ÷ (s ÷ √n), df = n − 1; Use when σ is unknown. If σ is known, use z = (x̄ − μ0) ÷ (σ ÷ √n).
Paired comparison: t = (d̄ − μd0) ÷ (sd ÷ √n), df = n − 1; d̄ is the mean of the paired differences and sd their standard deviation. n is the number of pairs. μd0 is usually 0.
Difference in means, independent samples, equal variances assumed: sp² = [(n1 − 1)s1² + (n2 − 1)s2²] ÷ (n1 + n2 − 2); t = [(x̄1 − x̄2) − (μ1 − μ2)0] ÷ √(sp²/n1 + sp²/n2); df = n1 + n2 − 2; Both populations assumed normal. If variances are not assumed equal, use √(s1²/n1 + s2²/n2) in the denominator, with df approximated.
Chi-square test of one variance: χ² = (n − 1)s² ÷ σ0², df = n − 1; σ0² is the hypothesised variance. The distribution is skewed, so two-tailed tests use two different critical values.
F-test of two variances: F = s1² ÷ s2², df = n1 − 1 and n2 − 1; Put the larger sample variance on top, so F ≥ 1. For a two-tailed test at significance α, use the α/2 table value.
Test of correlation (H0: ρ = 0): t = r√(n − 2) ÷ √(1 − r²), df = n − 2; Reject H0 if |t| exceeds the critical value. Assumes the two variables are approximately bivariate normal.
Spearman rank correlation: rs = 1 − [6 Σdi²] ÷ [n(n² − 1)]; di is the difference between the two ranks of observation i. n is the number of observations. Tied values get the average of their ranks (this formula is exact only without ties).
Test statistic for Spearman correlation: t = rs × √(n − 2) ÷ √(1 − rs²), with n − 2 degrees of freedom; Same t-statistic form as the parametric correlation test. Reject H0 of zero correlation if |t| exceeds the critical value.
Chi-square test of independence: χ² = Σ (Oij − Eij)² ÷ Eij; Oij is the observed count in a cell, Eij the expected count if the variables are independent.
Expected frequency in a cell: Eij = (row i total × column j total) ÷ grand total; Computed under the null hypothesis that the two variables are independent.
Degrees of freedom for independence test: df = (r − 1)(c − 1); r is the number of rows and c the number of columns. The test is one-sided (right tail).
Sign test idea: Count + and − signs; under H0 each sign has probability 0.5; The number of positive signs follows a binomial distribution with p = 0.5. Zero differences are dropped from n.

Quick revision

Standard error of the sample mean = s ÷ √n (or σ ÷ √n when σ is known).
The central limit theorem: for large samples, the distribution of the sample mean is approximately normal, whatever the population shape, given finite variance.
Confidence interval = point estimate ± (reliable factor × standard error).
Use z when the population variance is known; use t when it is unknown, with n − 1 degrees of freedom for a mean.
A higher confidence level or a smaller sample gives a wider interval.
Type I error means rejecting a true null; its probability is the significance level. Type II error means failing to reject a false null.
Power = 1 − probability of a Type II error.
Reject H0 if the test statistic falls in the rejection region, or if the p-value is below the significance level.
Test statistic = (sample statistic − hypothesized value) ÷ standard error.
Chi-square tests a single variance; the F-test compares two variances. In a two-tailed F-test, the larger sample variance is conventionally placed in the numerator and the upper critical value is used.
Bootstrap resamples with replacement; jackknife leaves out one observation at a time.
Use nonparametric tests when assumptions fail, data are ranks, or outliers dominate.

Common mistakes

Confusing stratified and cluster sampling. Fix: Ask whether every group is sampled. Stratified samples every stratum. Cluster samples only the randomly chosen clusters.
Calling sampling error a bias. Fix: Sampling error is random chance and shrinks with larger samples. Bias is systematic and does not go away by increasing sample size.
Using the standard deviation instead of the standard error when working with a sample mean. Fix: If the question is about x̄, always divide σ by √n first.
Dividing by n instead of √n. Fix: Standard error uses √n. Variance of the mean uses n.
Saying the jackknife samples with replacement. Fix: Jackknife leaves one out and has no repeats. Replacement belongs to bootstrap.
Thinking jackknife resamples have size n. Fix: Each jackknife sample has n − 1 observations, and there are n of them.
Saying 'accept the null hypothesis' when the test fails to reject. Fix: Failing to reject means the evidence was not strong enough against H0. It does not prove H0. Use 'fail to reject'.
Putting the claim to be proven in H0 or leaving the equality out of H0. Fix: H0 always has =, ≤ or ≥. Put the effect you hope to demonstrate in Ha.
Using an independent-samples test on paired data. Fix: If the same units are measured twice, or the observations are matched, work with the differences and use n pairs and n − 1 df.
Using n instead of n − 2 for the correlation test. Fix: Correlation uses n − 2 in both the formula and the df, because two sample means are estimated, one for each variable, so df = n − 2.

Exam tips

Memorize one clue word per method and bias. Most questions are direct matches.
For stratified versus cluster, the deciding test is whether all groups are sampled.
Survivorship bias nearly always makes performance look better and risk look lower.
Remember that more data reduces sampling error but does not cure bias.
With no penalty for wrong answers, always answer. If the stem states that selection chances are known, rule out convenience and judgmental sampling, then use the group-sampling test to separate stratified, cluster, systematic and simple random.
Look for the phrase 'sample mean'. It signals that you need the standard error, not the standard deviation.
The wrong options often include the raw standard deviation. Compute √n first and eliminate it.
For 'what happens if n increases' questions, answer with direction and the square root relationship. No calculation is needed.