Lesson 6: Hypothesis Testing & Correlation

📹 Video Overview

📊 What We're Learning

1. Hypothesis Testing - How to test if claims about data are true

2. Relationships Between Variables - How two things are connected

3. Correlation & Covariance - Measuring the strength of relationships

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🎯 PART 1: HYPOTHESIS TESTING

The Big Picture

The Problem: We can't check EVERYONE in a population, so we use a sample. But samples have errors!

The Solution: Design a test that minimizes the chance of making wrong decisions.

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🏛️ The Court Analogy (Best Way to Understand!)

Think of hypothesis testing like a court trial:

The Two Hypotheses

Hypothesis	Symbol	Meaning	Court Example
Null Hypothesis	H₀	The "default" assumption	Person is innocent
Alternative Hypothesis	H₁	What we're trying to prove	Person is guilty

Memory Hack: "Innocent Until Proven Guilty"

• H₀ = Status quo (nothing special happening)
• H₁ = The exciting claim (something IS happening)

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⚠️ Two Types of Errors

Type I Error (α - Alpha)

What it is: Rejecting H₀ when it's actually TRUE

Memory Hack: "False Alarm" 🚨

• Like sending an innocent person to jail
• We SET this probability ourselves (usually α = 0.05 or 5%)

Example: Saying a medicine works when it actually doesn't

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Type II Error (β - Beta)

What it is: NOT rejecting H₀ when it's actually FALSE

Memory Hack: "Missed Detection" 🙈

• Like letting a guilty person go free
• Harder to control directly

Example: Saying a medicine doesn't work when it actually does

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Error Comparison Table

Reality → Decision ↓	H₀ is TRUE	H₁ is TRUE
Don't Reject H₀	✓ Correct	✗ Type II Error (β)
Reject H₀	✗ Type I Error (α)	✓ Correct (Power = 1-β)

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🔢 Key Terms Explained Simply

Level of Significance (α)

• The probability of Type I error we're willing to accept
• Usually 0.05 (5%) or 0.01 (1%)
• Memory Hack: "How much risk of false alarm we tolerate"

Power of Test (1 - β)

• Probability of correctly rejecting H₀ when it's false
• Memory Hack: "How good our test is at catching the truth"

Critical Value

• The dividing line: reject H₀ on one side, don't reject on the other
• Memory Hack: "The fence between guilty and not guilty"

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📋 The 5 Steps of Hypothesis Testing

Step-by-Step Example: Do Men and Women Earn Different Salaries?

Step 1: State Hypotheses

• H₀: W(m) = W(f) → Salaries are the SAME
• H₁: W(m) ≠ W(f) → Salaries are DIFFERENT

Step 2: Choose α

• Let's use α = 0.05 (5% chance of false alarm)

Step 3: Pick Test Statistic

• We use two-sample t-test

Step 4: Calculate T-statistic

T = Signal / Noise = Difference between groups / Variability of groups

T = |x̄₁ - x̄₂| / √(s₁²/n₁ + s₂²/n₂)

Memory Hack for T-Test: "Signal vs Noise Radio"

• Signal (numerator): How different are the group averages?
• Noise (denominator): How much do the groups vary internally?
• Big T = Strong signal → Groups ARE different!
• Small T = Weak signal → Can't tell them apart from noise

Step 5: Make Decision

• If T is big enough → Reject H₀ (salaries ARE different!)
• If T is small → Don't reject H₀ (not enough evidence)

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🎯 PART 2: RELATIONSHIPS BETWEEN VARIABLES

What is a Statistical Relationship?

Simple Definition: When one variable changes, the other one changes too (in a predictable way)

IMPORTANT: Statistical relationship ≠ Causal relationship!

Memory Hack: "Ice Cream and Drowning"

• Ice cream sales and drowning deaths are correlated
• But ice cream doesn't CAUSE drowning!
• Both happen more in summer (hidden variable)

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📈 Scatter Plots (The Visual Way)

Example: Experience vs Salary

Employee	Experience (x)	Salary (y)
1	2	7,000
2	4	10,000
3	5	8,000
4	7	11,000
5	8	13,000
6	9	15,000
7	12	13,000
8	14	16,000
9	20	17,000
10	25	19,000

When you plot this, you see points going UP-RIGHT → positive relationship!

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🎨 Reading Scatter Plots

Visual Guide

Pattern	Relationship	Example
↗️ Tight line	Perfect positive (r = 1)	Age vs Height (kids)
↗️ Scattered up	Weak positive (r ≈ 0.3)	Study time vs Grade
↘️ Tight line	Perfect negative (r = -1)	Gas in tank vs Distance driven
↘️ Scattered down	Weak negative (r ≈ -0.3)	TV time vs Grade
🌐 Random	No relationship (r = 0)	Shoe size vs IQ
📈 Curve	Non-linear (r = 0)	Age vs Reaction time

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🔗 Covariance

What It Measures

How two variables move TOGETHER (but in their original units)

Formula

cov(x,y) = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / n

Memory Hack: "The Direction Detector"

• cov(x,y) > 0 → Positive relationship (both increase together)
• cov(x,y) < 0 → Negative relationship (one up, one down)
• cov(x,y) = 0 → No linear relationship

The Problem with Covariance

It depends on the units! Hard to interpret and compare.

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⭐ Pearson Correlation Coefficient (r)

What It Is

Standardized covariance → Same as covariance but on a scale of -1 to 1

Formula

r(x,y) = cov(x,y) / (sₓ × sᵧ)

Memory Hack: "Covariance with Training Wheels"

• Takes covariance
• Divides by both standard deviations
• Now it's always between -1 and 1!

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🎯 Interpreting Correlation (r)

The Scale

-1 ←------- 0 -------→ +1

Perfect    No      Perfect
Negative  Relationship  Positive

Direction (Look at Sign)

• r > 0 → Positive relationship ↗️
• r < 0 → Negative relationship ↘️
• r = 0 → No LINEAR relationship

Strength (Look at Absolute Value)

| |r| value | Strength | Interpretation | |-----------|----------|----------------| | 0.00 - 0.10 | Negligible | Basically no relationship | | 0.10 - 0.39 | Weak | Slight tendency | | 0.40 - 0.69 | Medium | Noticeable pattern | | 0.70 - 0.89 | Strong | Clear relationship | | 0.90 - 1.00 | Very Strong | Almost perfect |

Memory Hack: "The Absolute Rule"

• r = 0.85 → STRONG positive
• r = -0.85 → STRONG negative (same strength, opposite direction!)

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📊 Complete Example: Stock Returns

Given Data

Day	Stock A (x)	Stock B (y)
1	1.0%	3.0%
2	1.5%	4.5%
3	2.2%	4.7%
4	1.4%	4.0%
5	0.2%	3.5%

Step 1: Calculate Means

• x̄ = (1.0 + 1.5 + 2.2 + 1.4 + 0.2) / 5 = 1.26%
• ȳ = (3.0 + 4.5 + 4.7 + 4.0 + 3.5) / 5 = 3.94%

Step 2: Calculate Covariance

cov(x,y) = [(1.0-1.26)(3.0-3.94) + (1.5-1.26)(4.5-3.94) + ... + (0.2-1.26)(3.5-3.94)] / 5

cov(x,y) = 0.31

Interpretation: Positive covariance → stocks tend to move together!

Step 3: Calculate Standard Deviations

• sₓ = 0.66
• sᵧ = 0.62

Step 4: Calculate Correlation

r(x,y) = 0.31 / (0.66 × 0.62) = 0.31 / 0.41 = 0.76

Interpretation: r = 0.76 → STRONG positive correlation

• When Stock A goes up, Stock B tends to go up too!

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💰 BONUS: Portfolio Risk & Return

Key Concepts

Return Formula

Return = (Price(t+1) + Dividend - Price(t)) / Price(t)

Memory Hack: "What you gained divided by what you paid"

Variance of a SUM (IMPORTANT!)

var(x + y) = var(x) + var(y) + 2×cov(x,y)

Why This Matters for Investing

Memory Hack: "Don't Put All Eggs in One Basket"

If you invest in two stocks:

• Positive correlation: They move together → more risk!
• Negative correlation: They move opposite → LESS risk! (diversification)

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📝 Real Example: Portfolio Calculation

Scenario: WAR, RECESSION, STABLE, PROSPERITY, PEACE

State	Prob	Stock C Return	Stock D Return
WAR	15%	67%	-60%
RECESSION	25%	-20%	-40%
STABLE	35%	7%	13%
PROSPERITY	15%	27%	67%
PEACE	10%	-33%	233%

Calculate Expected Returns

Stock C:

E(rC) = 15%×67% + 25%×(-20%) + 35%×7% + 15%×27% + 10%×(-33%)
E(rC) = 8%

Stock D:

E(rD) = 15%×(-60%) + 25%×(-40%) + 35%×13% + 15%×67% + 10%×233%
E(rD) = 19%

Calculate Variance

VAR(rC) = 15%×(67%-8%)² + ... + 10%×(-33%-8%)²
VAR(rC) = 0.0069

VAR(rD) = 0.0152

Calculate Covariance

COV(rC, rD) = 15%×(67%-8%)×(-60%-19%) + ... + 10%×(-33%-8%)×(233%-19%)
COV(rC, rD) = -0.00176

Negative covariance = GOOD for portfolio! Stocks move in opposite directions.

Calculate Correlation

ρ = -0.00176 / √(0.0069 × 0.0152)
ρ = -0.171

Interpretation: Weak negative correlation → some diversification benefit!

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🧠 Memory Hacks Summary

Concept	Memory Trick
Hypothesis Testing	"Court trial - innocent until proven guilty"
Type I Error	"False alarm - innocent in jail"
Type II Error	"Missed detection - guilty goes free"
Alpha (α)	"How much false alarm risk we accept"
T-test	"Signal vs Noise radio"
Covariance	"Direction detector (with units)"
Correlation	"Covariance with training wheels (-1 to 1)"
Portfolio Risk	"Don't put all eggs in one basket"

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🎯 Quick Reference Formulas

Two-Sample T-Test:
T = |x̄₁ - x̄₂| / √(s₁²/n₁ + s₂²/n₂)

Covariance:
cov(x,y) = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / n

Pearson Correlation:
r(x,y) = cov(x,y) / (sₓ × sᵧ)

Variance of Sum:
var(x + y) = var(x) + var(y) + 2×cov(x,y)

Return:
Return = (Price(t+1) + Dividend - Price(t)) / Price(t)

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📊 Decision Flow Chart

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💡 Pro Tips for Exams

1. Hypothesis Testing:

   • Always start with H₀ (null)
   • H₁ is what you're trying to prove
   • Be careful: "Don't reject" ≠ "Accept"

2. Correlation:

   • Sign shows direction (+ or -)
   • Absolute value shows strength
   • r = 0 means NO LINEAR relationship (might be curved!)

3. Common Mistakes:

   • Don't confuse correlation with causation
   • Don't forget to take square root when going from variance to SD
   • Remember covariance has units, correlation doesn't

4. Calculator Check:

   • Correlation must be between -1 and 1
   • If you get r = 2.5, you made a mistake!

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Good luck! Remember: Statistics is about making the best decision with incomplete information! 🎲📊