Estimation and Hypothesis Testing

Analysts rarely observe a whole population; they work with samples and must judge how far a sample statistic can be trusted. This module covers how samples are drawn and how they can be biased, why the sample mean is approximately normal in large samples, how to build confidence intervals, how to run and interpret a hypothesis test, and when a non-parametric test is the better tool.

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What you will learn

1. Compare probability and non-probability sampling methods, explain sampling error, and identify common biases in samples of financial data.

Probability sampling: simple random, systematic, stratified and cluster; non-probability: convenience and judgmental.
Stratified sampling ensures each subgroup is represented; it is common in bond index replication.
Sampling error = sample statistic - population parameter; it shrinks as the sample grows.
Survivorship, look-ahead, data-snooping, time-period and self-selection biases distort financial samples.
Biases are systematic: a larger sample does not remove them.

2. Explain the central limit theorem and calculate and interpret the standard error of the sample mean.

The sampling distribution of the mean is centred on the population mean.
Standard error = sigma / square root of n, or s / square root of n when sigma is unknown.
Quadrupling the sample size halves the standard error.
CLT: for large n (rule of thumb 30 or more), the sample mean is approximately normal whatever the population's shape.
The CLT applies to the mean, not to individual returns, and assumes independent observations.

3. Construct and interpret confidence intervals for a population mean, choosing between the z- and t-distributions.

Confidence interval = point estimate +/- reliability factor x standard error.
Known variance: z-values 1.645, 1.96, 2.576 for 90%, 95%, 99%.
Unknown variance: t-values with n - 1 degrees of freedom; larger than z in small samples.
Higher confidence, more volatility and smaller samples all widen the interval.
Good estimators are unbiased, efficient and consistent.

4. Explain the steps and components of a hypothesis test, including the significance level, p-value, Type I and Type II errors and power, and test a hypothesis about a population mean.

The null hypothesis contains the equality; the alternative is what the analyst seeks to support.
Test statistic for a mean: t = (sample mean - hypothesized mean) / (s / square root of n), df = n - 1.
Reject the null if the statistic exceeds the critical value, or if the p-value is below alpha.
Type I error: reject a true null (probability alpha); Type II: fail to reject a false null (beta); power = 1 - beta.
Lowering alpha raises Type II risk; testing many hypotheses inflates false discoveries.

5. Compare parametric and non-parametric tests, identify when each is appropriate, and conduct tests of a correlation and of independence.

Parametric tests concern parameters and assume a distribution; non-parametric tests need fewer assumptions.
Use non-parametric tests for ranked data, outliers or small non-normal samples, or hypotheses not about parameters.
Correlation test: t = r x square root of (n - 2) / square root of (1 - r^2), df = n - 2.
Spearman rank correlation is the non-parametric alternative, robust to outliers.
Chi-square test of independence: expected count = row total x column total / grand total; df = (r - 1)(c - 1).

Module at a glance

Stratified random sampling guarantees subgroup representation; non-probability samples may be unrepresentative.
Survivorship, look-ahead, data-snooping, time-period and self-selection biases are not cured by more data.
Standard error = s / square root of n; the CLT makes the sample mean approximately normal for large n.
Confidence interval = estimate +/- reliability factor x standard error; use t with n - 1 df when the variance is estimated.
Reject the null when the test statistic is beyond the critical value or the p-value is below alpha.
Type I: reject a true null (alpha); Type II: miss a false null (beta); power = 1 - beta.
Use non-parametric tests (Spearman, chi-square independence) for ranks, outliers or non-parameter hypotheses.

This free module has 5 lessons, 13 practice questions with full explanations.

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