Quantitative Methods and Statistics
Quantitative Methods and Statistics tests probability, common distributions, sampling, confidence intervals, and hypothesis testing. The exam emphasizes correct interpretation of standard errors, p-values, and interval estimates rather than memorized output.
Core rules and concepts
- Expected value: E(X) = Σ x·P(x). Variance is E[(X − μ)²]. For independent variables, Var(X + Y) = Var(X) + Var(Y).
- Normal distribution: about 68% of observations lie within one standard deviation, 95% within two, and 99.7% within three. Use the t distribution when the population standard deviation is unknown and the sample is small.
- Sampling distribution: the standard error of a mean is σ/√n, estimated by s/√n. The central limit theorem makes the sample mean approximately normal for large n.
- Confidence interval: x̄ ± z*·SE for a known σ, with z* = 1.96 for 95% confidence. The interval either contains the fixed parameter or it does not; it does not give a 95% probability to the parameter after the sample is drawn.
- Hypothesis testing: specify H0 and H1 before seeing the data. α is the Type I error rate; β is the Type II error rate. A p-value is the probability of a result at least as extreme as observed, assuming H0 is true.
Worked example
A sample of n = 100 has mean 52 and known σ = 10. The standard error is 10/√100 = 1. The 95% confidence interval is 52 ± 1.96 × 1, or [50.04, 53.96]. To test H0: μ = 50 against H1: μ ≠ 50, compute z = (52 − 50)/1 = 2.00. The two-sided p-value is approximately 0.0455, so reject H0 at the 5% level but not at the 1% level. The reasoning combines the standard error, the test statistic, and the two-tailed comparison.
Traps that cost marks
The standard deviation describes individual observations; the standard error describes a sample statistic. A p-value is not the probability that H0 is true. Do not switch between one-tailed and two-tailed tests after seeing the direction of the result. Use t rather than z when σ is estimated from a small sample, and remember that statistical significance is not the same as economic importance.