CFA Level I · CFA Level I Exam

Estimation and Hypothesis Testing for CFA Level I

Estimation and hypothesis testing is how you use a sample to learn about a population. You estimate a parameter with a point estimate and confidence interval, then test a claim by comparing a test statistic with a critical value or p-value. Decide whether to reject the null hypothesis.

What this chapter covers

This chapter covers statistical inference. You rarely see a whole population, so you draw a sample and use it to say something about the population. The chapter starts with how samples are drawn and where bias creeps in. It then moves to the central limit theorem, standard error, point estimates and confidence intervals. Resampling methods (bootstrap and jackknife) show how to estimate the spread of a statistic without a formula.

The second half is hypothesis testing. You learn the framework: state the null and alternative, choose a significance level, compute a test statistic, and decide. Then you apply it to means, variances and correlation, using z, t, chi-square and F tests. The chapter ends with parametric versus nonparametric tests, which you use when the usual assumptions fail.

The chapter sits on top of probability and sampling distributions, and it feeds directly into what follows. Regression, time-series work, and performance evaluation all rely on t-tests and confidence intervals. In Portfolio Construction, Equities and Fixed Income, claims about returns or risk are often judged by significance. Ethics-style wording about data mining and backtesting also links to sampling bias.

Quantitative Methods carries a meaningful share of the 180 questions, and this chapter supplies many of its most testable, rule-based items. Questions are standalone and three-option, so they reward clean logic: pick the right test, the right tail, the right degrees of freedom, then decide. Each of these is a skill you can master with practice. The same ideas also appear in regression and in other topics, so time spent here pays off beyond one chapter. There is no penalty for wrong answers, but you can often eliminate two options by checking direction and size, which makes careful practice worth the effort.

Estimation and Hypothesis Testing: topics in the order to study them

1Sampling Methods and Sampling BiasStart here because every later idea assumes you know how a sample is drawn and what can go wrong with it.
2Central Limit Theorem and Standard ErrorThis explains why sample means behave predictably, and it is the base for confidence intervals and test statistics.
3Point Estimates and Confidence IntervalsOnce you know standard error, you can build intervals and choose between z and t.
4Resampling: Bootstrap and JackknifeA short, concept-based topic that extends estimation; it is easier once you understand standard error.
5Hypothesis Testing FrameworkYou need the steps, error types and decision rules before applying any specific test.
6Tests of Means, Variances and CorrelationThis is the calculation-heavy core, and it builds directly on the framework and on intervals.
7Parametric vs Nonparametric TestsStudy this last, since it asks when the earlier tests are not suitable and what to use instead.

How to prepare Estimation and Hypothesis Testing

Aim for understanding first, then speed. Most questions can be solved in about 90 seconds once the pattern is clear.

Read each topic once for the idea, and write one sentence on what problem it solves.
Learn the formulas for standard error, confidence interval and test statistic, and say aloud when each one applies.
Build a decision table on one page: which test (z, t, chi-square, F) for which question, with degrees of freedom.
Do the calculations with your approved calculator (TI BA II Plus or HP 12C) so square roots and divisions are quick and reliable.
Practise setting up hypotheses before computing anything: write H0 and Ha, decide one-tailed or two-tailed, then find the critical value.
Do mixed sets of standalone three-option questions, and for each wrong answer note whether the error was the test choice, the tail, or arithmetic.
In the last week, redo only your error log and the one-page decision table.

Common mistakes in Estimation and Hypothesis Testing

Using z when t is needed, or the reverse.
Fix: Ask first: is the population variance given? If not, use t with n − 1 degrees of freedom, even for fairly large samples.
Using the wrong tail or critical value.
Fix: Write Ha first. If it says greater than or less than, use one tail. If it says not equal, split the significance level between two tails.
Confusing standard deviation with standard error.
Fix: Standard deviation describes the data. Standard error describes the sample statistic, and for a mean it is divided by √n.
Saying you 'accept' the null hypothesis.
Fix: Use 'fail to reject'. A test can lack evidence against H0 without proving it.
Mixing up Type I and Type II errors, or misreading the p-value.
Fix: Type I is rejecting a true null and equals the significance level. The p-value is the smallest significance level at which you would reject H0, not the probability that H0 is true.
Ignoring sampling bias and data mining in scenario questions.
Fix: Scan for survivorship, self-selection, look-ahead and time-period bias before computing. A precise estimate from a biased sample is still unreliable.

Last-day revision: Estimation and Hypothesis Testing

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.

Estimation and Hypothesis Testing practice questions

Estimation and Hypothesis Testing in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Estimation and Hypothesis Testing: frequently asked questions

How do I decide between a z-test and a t-test?

Check whether the population variance is known. If it is, use z. If it is unknown and you use the sample standard deviation, use t with n − 1 degrees of freedom for a mean. For a large sample the two give close results, but the t-test is still the correct choice when the variance is unknown.

What is the difference between a confidence interval and a hypothesis test?

A confidence interval gives a range of plausible values for a parameter. A hypothesis test checks a specific claim about it. They are linked: for a two-tailed test, you reject H0 at the matching significance level if the hypothesized value lies outside the confidence interval.

Do I need to memorise many critical values?

Learn the common z values, such as 1.645 for 90%, 1.96 for 95% and 2.576 for 99% two-tailed intervals. Questions generally supply t, chi-square and F values when they are needed, so focus on knowing which one to pick and how to find the degrees of freedom.

How are bootstrap and jackknife different?

The bootstrap draws many random samples with replacement from the original sample and computes the statistic each time. The jackknife removes one observation at a time and recomputes the statistic. Both estimate the spread of a statistic without relying on a formula.