Hypothesis Testing

4. Hypothesis Testing — Making Decisions with Data

4.1 The Framework

🧠 Court analogy se samjho: H₀ = "Accused is innocent" (default). H₁ = "Accused is guilty" (claim). Evidence = Data. If evidence itna strong hai ki innocence mein possible nahi (p < 0.05) → Guilty! If evidence weak hai → Not Guilty.

4.2 p-value Explained

p-value = "If H₀ is true, what is the probability of observing results this extreme or more?"

• p = 0.03 → Only 3% chance this result is random → Pattern is REAL → Reject H₀
• p = 0.45 → 45% chance this is random → Nothing special → Fail to Reject H₀

Worked Problem — Business Hypothesis:

A marketing team claims their new email subject line has increased open rates from the old 22% to 27%.

H₀: Open rate = 22% (no change)
H₁: Open rate > 22% (improvement)

Data: 500 emails sent, 135 opened → Observed rate = 27%
Test: One-proportionz-test
Result: p-value = 0.012

Decision: p = 0.012 < α = 0.05 → REJECT H₀
Conclusion: The new subject line significantly improved open rates ✅

4.3 Type I vs Type II Errors

Error	What Happened	Analogy	Business Example
Type I (α)	Rejected H₀ when it was true	Convicting innocent	Concluding campaign worked when it didn't → wasted budget
Type II (β)	Failed to reject H₀ when H₁ was true	Letting guilty go free	Concluding campaign failed when it actually worked → missed opportunity

🧠 Yaad kaise rakho: Type I = "I wrongly accused" (innocent ko pakda). Type II = criminal "eIIude" kiya (guilty bach gaya).

4.4 Statistical Power

Power = 1 - β = Probability of correctly rejecting H₀ when H₁ is true.

• Standard target: 80% power (β = 0.20)
• Higher power = larger sample size needed

Factors that increase power:

1. Larger sample size
2. Larger effect size
3. Higher significance level (α)
4. Lower variability in data

4.5 Common Statistical Tests

Test	When to Use	Example	Data Type
One-sample t-test	Compare sample mean to known value	"Is our avg delivery time different from 3 days?"	1 continuous variable
Two-sample t-test	Compare means of 2 groups	"Are Delhi and Mumbai avg order values different?"	1 continuous + 1 categorical (2 groups)
Paired t-test	Compare same group at 2 times	"Did training improve employee scores?"	2 paired continuous measurements
Chi-Square	Test association between categorical variables	"Is gender linked to product preference?"	2 categorical variables
ANOVA	Compare means across 3+ groups	"Are revenues different across cities?"	1 continuous + 1 categorical (3+ groups)
Pearson Correlation	Linear relationship between 2 continuous vars	"Is there a link between ad spend and sales?"	2 continuous variables

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