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.

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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

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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

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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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