Σ Quantitative Methods

Hypothesis Testing — Steps, T-Test, Type I & Type II Errors

Hypothesis testing explained step by step. Null vs alternative hypothesis, t-test, z-test, p-value, Type I and Type II errors with examples.

Key Concepts

Null & Alternative Hypotheses

H<sub>0</sub>: status quo (what we try to reject). H<sub>a</sub>: what we want to prove. We never "accept" H<sub>0</sub>, only fail to reject.

Type I and Type II Errors

Type I (α): reject true H<sub>0</sub> (false positive). Type II (β): fail to reject false H<sub>0</sub> (false negative). Power = 1 - β.

Test Statistic

test stat = (sample stat - hypothesized value) / standard error. Compare to critical value.

p-value

Smallest significance level at which H<sub>0</sub> would be rejected. Reject H<sub>0</sub> if p-value < α.

One-tailed vs Two-tailed

Two-tailed: H<sub>a</sub>: μ ≠ value. One-tailed: H<sub>a</sub>: μ > value or μ < value. Two-tailed has critical values on both sides.

Formulas

From this module

Test Statistic (mean)

t = (X̄ - μ0) / (s / √n)

Where: μ<sub>0</sub> = hypothesized mean

Test for Difference in Means

t = (X̄1 - X̄2) / √(s²1/n1 + s²2/n2)

Where: For independent samples

F-test for Variance Equality

F = s²1 / s²2 (larger variance in numerator)

Where: df<sub>1</sub> = n<sub>1</sub>-1, df<sub>2</sub> = n<sub>2</sub>-1

Master Formula Sheet -- Quantitative Methods

Future Value

FV = PV × (1 + r)n

Single lump sum compounding

Present Value

PV = FV / (1 + r)n

Discounting future cash flows

Annuity PV

PV = PMT × [1 - (1+r)-n] / r

Equal periodic payments

Effective Annual Rate

EAR = (1 + r/m)m - 1

m = compounding periods per year

Holding Period Return

HPR = (P₁ - P₀ + D) / P₀

Total return including income

Population Variance

σ² = Σ(Xᵢ - μ)² / N

Divide by N for population

Sample Variance

s² = Σ(Xᵢ - X̄)² / (n-1)

Divide by n-1 for sample (Bessel's correction)

Coefficient of Variation

CV = σ / μ

Risk per unit of return

Sharpe Ratio

Sharpe = (R̄ₚ - Rᶠ) / σₚ

Excess return per unit of total risk

Bayes' Formula

P(A|B) = P(B|A) × P(A) / P(B)

Updating probabilities with new info

Correlation

ρ = Cov(X,Y) / (σₓ × σᵧ)

Standardized covariance, -1 to +1

Linear Regression

Y = b₀ + b₁X + ε

b₁ = Cov(X,Y)/Var(X)

Confidence Interval

X̄ ± z(α/2) × (σ/√n)

z = 1.96 for 95% CI

Test Statistic

t = (X̄ - μ₀) / (s/√n)

Compare to critical value

Decision Frameworks

Which test to use?

Use when:

  • t-test: testing means (unknown σ)
  • F-test: testing equality of two variances
  • Chi-square: testing a single variance, contingency tables

Avoid when:

  • Using z-test when σ is unknown and sample is small

Test Your Understanding

A Type I error occurs when a researcher:

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