A portfolio manager claims her strategy earns a positive return. You have 36 monthly returns. Estimation tells you what the true mean probably is; hypothesis testing tells you whether the claim survives the data.
You almost never see a whole population. You see a sample, compute a statistic like the sample mean, and use it to estimate the population parameter. The gap between the two is sampling error. Good estimation is about keeping that error small and knowing how big it is.
How you draw the sample decides whether it represents the population. Probability sampling gives every member a known chance of selection. Non-probability sampling relies on convenience or judgment and is cheaper but riskier.
Stratified sampling is the one the exam likes. A bond index fund that samples each rating and maturity cell in proportion usually tracks the index more closely than a simple random sample of the same size, because every cell gets its right share. It lowers the expected tracking error; it does not guarantee a smaller error in every sample.
Common mistakes
- Putting the claim in the null. The null always carries the equality. "The mean exceeds zero" is , never .
- Mixing up one-tailed and two-tailed critical values. In Example 1, comparing t = 2.00 with 2.030 leads to "fail to reject". Trap: the question asked whether the mean exceeds zero, so the right critical value is 1.690 and you reject.
- Testing paired data as independent samples. In Example 3 the independent-samples t is 0.24, which never rejects. Trap: the paired t of 2.50 is the correct statistic.
Bottom line
- The CLT makes the sample mean approximately normal for large n (rule of thumb n ≥ 30). Standard error = s / √n.
- Confidence interval = estimate ± reliability factor × standard error. Use z with known variance, t with n − 1 df otherwise.
- The null always holds the equality; a claim you want to prove goes in .
- Type I error = reject a true null (α). Type II = miss a false null (β). Power = 1 − β. A larger sample or a less noisy design cuts both errors.
Exam shortcut
Read the alternative first. It tells you whether the test is one-tailed or two-tailed, and so which critical value to use. Then match the data to the test: same months, use paired; a stated target variance, use chi-square; two variances, use F with the larger on top; ranks, use Spearman; counts in a table, use chi-square with (r − 1)(c − 1) df. Memorize 1.645, 1.96 and 2.576 for z.
The full lesson (about 4,171 words, 28 min read) adds 5 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- estimation and hypothesis testing
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