Chapter 3: Introduction to Hypothesis Testing

Welcome to Hypothesis Testing! 🎯

Hypothesis testing is one of the most powerful tools in statistics. It provides a rigorous, mathematical framework for making decisions when we’re uncertain about the truth.

The Core Challenge:

Every day, people make claims:

• “Our new app increases user engagement by 20%”
• “This teaching method improves test scores”
• “Men and women have different preferences for this product”
• “The average delivery time is under 30 minutes”

But how do we know if these claims are true or just random chance? That’s where hypothesis testing comes in.

The Detective Analogy:

Imagine you’re a detective investigating a case:

• The Suspect: A claim about the world (hypothesis)
• Your Job: Collect evidence (data) to determine if the claim is likely true or false
• The Standard: “Beyond reasonable doubt” (statistical significance)
• The Verdict: Reject the claim or fail to reject it based on evidence

Just like a detective doesn’t “prove” guilt but shows evidence beyond reasonable doubt, we don’t “prove” hypotheses - we determine if there’s enough statistical evidence to support or reject them.

Table of Contents

1. What is Hypothesis Testing?
2. The Detective’s Toolkit - Key Concepts
3. The 5-Step Process
4. Types of Errors
5. One-Tailed vs Two-Tailed Tests
6. Complete Python Example

1. What is Hypothesis Testing?

The Simple Explanation

Hypothesis testing is like being a judge in a courtroom:

• Someone makes a claim (like “This coin is fair”)
• You collect evidence (flip the coin many times)
• You decide: Is there enough evidence to reject the claim?

Real-Life Examples

Example 1: The Coin Mystery 🪙

• Claim: “This coin is fair (50% heads, 50% tails)”
• Evidence: You flip it 100 times and get 65 heads
• Question: Is the coin really fair, or is it biased?

Example 2: The Medicine Mystery 💊

• Claim: “This new medicine works better than the old one”
• Evidence: You test it on 200 patients
• Question: Does the new medicine really work better?

Example 3: The Website Mystery 🖱️

• Claim: “The new website design gets more clicks”
• Evidence: You track clicks for 1000 visitors
• Question: Is the new design really better?

2. The Detective’s Toolkit - Key Concepts

🎯 The Two Hypotheses

Every hypothesis test has TWO competing claims:

H₀ (Null Hypothesis) - “The Boring Claim”

• This is what everyone currently believes
• It’s the “nothing special is happening” claim
• We try to find evidence AGAINST this

Examples:

• “The coin is fair” (50% heads)
• “The new medicine is no better than the old one”
• “The average height is 170 cm”

H₁ (Alternative Hypothesis) - “The Interesting Claim”

• This is what we think might be true
• It’s the “something special IS happening” claim
• We try to find evidence FOR this

Examples:

• “The coin is NOT fair” (not 50% heads)
• “The new medicine IS better”
• “The average height is NOT 170 cm”

📊 P-Value - “How Weird Is Your Evidence?”

The p-value is perhaps the most misunderstood concept in statistics. Let’s break it down clearly.

Formal Definition:

The p-value is the probability of observing data as extreme as (or more extreme than) what we actually observed, assuming the null hypothesis is true.

What does this really mean?

Imagine you flip a coin 100 times and get 65 heads. You wonder: “Is this coin fair?”

• If the coin IS fair (H₀ is true), what’s the chance of getting 65+ heads just by luck?
• That probability is the p-value
• If it’s very small (say, 0.01 or 1%), then either:
  • You witnessed a very rare event, OR
  • The coin isn’t actually fair (more likely!)

Interpreting P-Values:

• p < 0.01 (Very small): “This is extremely unlikely to happen by chance. Strong evidence against H₀”
• p < 0.05 (Small): “This is unlikely to happen by chance. Moderate evidence against H₀”
• p > 0.05 (Large): “This could easily happen by chance. Weak evidence against H₀”
• p > 0.10 (Very large): “This is quite likely to happen by chance. No evidence against H₀”

The Magic Number: 0.05 (5%)

By convention, we use α = 0.05 as our threshold:

• If p-value < 0.05: Evidence is strong enough → Reject H₀ (statistically significant)
• If p-value ≥ 0.05: Evidence is not strong enough → Fail to reject H₀ (not significant)

Important Misconceptions:

❌ WRONG: “p-value is the probability that H₀ is true” ✅ CORRECT: “p-value is the probability of seeing this data if H₀ were true”

❌ WRONG: “p = 0.05 means there’s a 5% chance we’re wrong” ✅ CORRECT: “p = 0.05 means if H₀ is true, we’d see data this extreme 5% of the time”

❌ WRONG: “p < 0.05 proves H₁ is true” ✅ CORRECT: “p < 0.05 suggests H₀ is unlikely, so we reject it in favor of H₁”

🧮 Understanding the Formulas - Built From Scratch

Before we dive into calculations, let’s truly understand what these formulas mean by building them up step by step!

🎯 The Core Idea: Measuring “Weirdness”

All hypothesis testing formulas answer one question:

“How weird is my result if H₀ is true?”

Think of it like this:

• You claim to be 6 feet tall
• I measure you and get 6 feet 1 inch
• Question: Is that difference (1 inch) a big deal or just measurement error?

The answer depends on:

1. How big the difference is (1 inch)
2. How much variation is normal (measuring tape accuracy)

This is exactly what our formulas do!

📏 Building Block 1: The Difference

Simple version:

Difference = What you got - What was claimed

Examples:

• Coin: Got 65% heads, claimed 50% → Difference = 15%
• Height: Got 171 cm, claimed 170 cm → Difference = 1 cm
• Wait time: Got 5.8 min, claimed 5.0 min → Difference = 0.8 min

In formulas:

• For proportions: p̂ - p₀ (your proportion - claimed proportion)
• For means: x̄ - μ₀ (your average - claimed average)

📏 Building Block 2: Standard Error (The “Normal Variation”)

What is Standard Error?

Standard Error (SE) tells you: “How much would my sample result typically vary just by random chance?”

Analogy: Imagine you’re measuring your height:

• If your measuring tape is accurate to ±0.1 cm → Small SE (precise!)
• If your measuring tape is accurate to ±5 cm → Large SE (imprecise!)

The Formula:

SE = "How spread out the data is" / √(Sample size)
     ↓                              ↓
   Variation                    More data = more precise

Why divide by √n?

• Bigger sample → More reliable → Smaller SE
• √n grows slower than n, which matches reality
• With 4× more data, you get 2× more precision (√4 = 2)

For proportions:

SE = √[p₀(1-p₀) / n]

Breaking it down:
- p₀(1-p₀) = Maximum variation for proportions
  - If p₀ = 0.5 (50%): p₀(1-p₀) = 0.5 × 0.5 = 0.25 (most variable)
  - If p₀ = 0.9 (90%): p₀(1-p₀) = 0.9 × 0.1 = 0.09 (less variable)
- Divide by n = Account for sample size
- √ = Take square root (statistical reasons)

For means:

SE = s / √n

Breaking it down:
- s = Sample standard deviation (how spread out your data is)
- √n = Square root of sample size
- Bigger s → Bigger SE (more variation)
- Bigger n → Smaller SE (more data = more precise)

Intuitive Example:

Imagine measuring wait times at a coffee shop:

• If wait times vary wildly (2 min to 20 min) → Large s → Large SE
• If wait times are consistent (4.5 min to 5.5 min) → Small s → Small SE
• If you only visit 4 times → Large SE (not enough data)
• If you visit 100 times → Small SE (lots of data!)

📏 Building Block 3: The Test Statistic (Putting It Together)

The Magic Formula:

Test Statistic = Difference / Standard Error
                 ↓           ↓
            How far off?   Normal variation?

What this means in plain English:

“My result is [test statistic] times more extreme than normal random variation”

Examples:

Example 1: Small test statistic (t = 0.5)

Difference = 0.8 minutes
SE = 1.6 minutes
t = 0.8 / 1.6 = 0.5

Interpretation: "My result is only 0.5 standard errors away from the claim.
That's well within normal random variation. Nothing unusual here!"

Example 2: Large test statistic (t = 3.0)

Difference = 0.8 minutes
SE = 0.27 minutes
t = 0.8 / 0.27 = 3.0

Interpretation: "My result is 3 standard errors away from the claim.
That's pretty far! This is unusual and probably not just random chance."

Visual Understanding:

Normal Random Variation (Standard Errors)

        -3    -2    -1     0     1     2     3
        |-----|-----|-----|-----|-----|-----|
        ↑                 ↑                 ↑
    Very unusual      Expected      Very unusual
    
If your test statistic falls here → Probably not random!
If your test statistic falls in middle → Could be random

Rule of Thumb:

• test statistic

  < 2: Probably just random variation

• test statistic

  ≈ 2: Borderline, might be real

• test statistic

  > 2: Probably a real difference!

🎓 Complete Formula Breakdown

Now let’s see the full formulas with everything labeled:

For Proportions (Z-test):

         What you got - What was claimed
z = ─────────────────────────────────────────
    √[Claimed × (1-Claimed) / Sample size]
    
         p̂ - p₀
  = ──────────────────
    √[p₀(1-p₀) / n]

Step by step:
1. Calculate difference: p̂ - p₀
2. Calculate SE: √[p₀(1-p₀) / n]
3. Divide: z = difference / SE

For Means (T-test):

         Your average - Claimed average
t = ─────────────────────────────────────
    Your std dev / √(Sample size)
    
         x̄ - μ₀
  = ──────────────
       s / √n

Step by step:
1. Calculate difference: x̄ - μ₀
2. Calculate SE: s / √n
3. Divide: t = difference / SE

🔢 Worked Example: Building Up the Calculation

Scenario: Coin flipped 100 times, got 65 heads. Is it fair?

Step-by-step with explanations:

Given:
- Flips (n) = 100
- Heads observed = 65
- Claimed probability (p₀) = 0.50 (fair coin)

Step 1: Calculate what you got
p̂ = 65/100 = 0.65 (you got 65% heads)

Step 2: Calculate the difference
Difference = p̂ - p₀ = 0.65 - 0.50 = 0.15
→ "You got 15 percentage points more heads than expected"

Step 3: Calculate the standard error (normal variation)
SE = √[p₀(1-p₀) / n]
SE = √[0.50 × 0.50 / 100]
SE = √[0.25 / 100]
SE = √0.0025
SE = 0.05
→ "With 100 flips of a fair coin, results typically vary by about 5%"

Step 4: Calculate the test statistic
z = Difference / SE
z = 0.15 / 0.05
z = 3.0
→ "Your result is 3 standard errors away from expected"
→ "This is 3× more extreme than normal random variation!"

Step 5: Interpret
z = 3.0 is pretty far from 0!
This is unusual enough to suspect the coin isn't fair.

🧮 How to Calculate P-Values - Step by Step

Now that you understand what the formulas mean, let’s see how to use them!

Example Calculation:

Coin flip example:
- Flipped 100 times, got 65 heads
- H₀: p = 0.50 (fair coin)
- p̂ = 65/100 = 0.65

Calculate z:
z = (0.65 - 0.50) / √[0.50(1-0.50)/100]
z = 0.15 / √[0.25/100]
z = 0.15 / √0.0025
z = 0.15 / 0.05
z = 3.0

Step 2: Convert Test Statistic to P-Value

The p-value is the probability of getting a test statistic as extreme (or more extreme) than what you observed, if H₀ is true.

Method 1: Using Z-Table or T-Table

For two-tailed test:

1. Find the area in the tail beyond your test statistic
2. Multiply by 2 (because we check both tails)

Example with z = 3.0:

1. Look up z = 3.0 in standard normal table
2. Area beyond z = 3.0 is 0.0013
3. Two-tailed p-value = 2 × 0.0013 = 0.0026

For one-tailed test:

• Just use the area in one tail (don’t multiply by 2)

Method 2: Using Python/Calculator

from scipy import stats

# For z-test (two-tailed)
z_statistic = 3.0
p_value = 2 * (1 - stats.norm.cdf(abs(z_statistic)))
print(f"p-value = {p_value:.4f}")  # Output: 0.0027

# For t-test (two-tailed)
t_statistic = 2.5
df = 29  # degrees of freedom (n-1)
p_value = 2 * (1 - stats.t.cdf(abs(t_statistic), df))
print(f"p-value = {p_value:.4f}")

Step 3: Interpret the P-Value

What the p-value tells you:

If p-value = 0.0027 (like our coin example):

• “If the coin were truly fair (H₀ true), there’s only a 0.27% chance of getting results as extreme as 65 heads out of 100 flips”
• This is very unlikely!
• Since 0.0027 < 0.05, we reject H₀

Visual Understanding:

Standard Normal Distribution (z-distribution)

                    |
                   /|\
                  / | \
                 /  |  \
                /   |   \
               /    |    \
              /     |     \
    ---------|-----|-----|----------
           -3.0    0    3.0
            ↑            ↑
         0.13%        0.13%
         
Total in both tails = 0.13% + 0.13% = 0.26% ≈ p-value

🎚️ Significance Level (α) - “How Sure Do We Need to Be?”

The significance level (alpha, α) is like setting the bar for how much evidence you need:

• Usually set to 0.05 (which means 95% confidence)
• Sometimes 0.01 (99% confidence - very strict!)
• Sometimes 0.10 (90% confidence - more lenient)

What it means:

• α = 0.05 means: “I’m willing to be wrong 5% of the time”
• It’s the threshold for deciding if evidence is strong enough

How α is Chosen

The significance level is chosen before you collect data based on:

1. Field Standards:
   • Science/Medicine: α = 0.05 or 0.01 (strict)
   • Social Sciences: α = 0.05 (standard)
   • Exploratory Research: α = 0.10 (lenient)
2. Consequences of Error:
   • High cost of Type I error → Use smaller α (0.01)

   • Example: Approving a dangerous drug

   • Lower cost of Type I error → Use larger α (0.10)
   • Example: Preliminary screening test
3. Trade-off:
   • Smaller α (0.01) → Harder to reject H₀ → Fewer Type I errors, more Type II errors
   • Larger α (0.10) → Easier to reject H₀ → More Type I errors, fewer Type II errors

Critical Values - The Decision Boundary

Instead of calculating p-values, you can use critical values to make decisions.

What is a critical value?

• The test statistic value that marks the boundary of the rejection region
• If your test statistic is more extreme than the critical value → Reject H₀

How to find critical values:

For Z-test:

Two-tailed test (α = 0.05):
Critical values: z = ±1.96

One-tailed test (α = 0.05):
Right-tailed: z = +1.645
Left-tailed: z = -1.645

Common Z critical values: | Confidence Level | α (two-tailed) | Critical Value | |——————|—————-|—————-| | 90% | 0.10 | ±1.645 | | 95% | 0.05 | ±1.96 | | 99% | 0.01 | ±2.576 |

For T-test:

• Critical values depend on degrees of freedom (df = n - 1)
• Look up in t-table or use Python

from scipy import stats

# Find critical t-value
alpha = 0.05
df = 29  # n - 1
critical_t = stats.t.ppf(1 - alpha/2, df)  # Two-tailed
print(f"Critical t-value: ±{critical_t:.3f}")
# Output: ±2.045

Decision Rules:

Method 1: P-value approach

If p-value < α → Reject H₀
If p-value ≥ α → Fail to reject H₀

Method 2: Critical value approach

Two-tailed:
If |test statistic| > critical value → Reject H₀
If |test statistic| ≤ critical value → Fail to reject H₀

One-tailed (right):
If test statistic > critical value → Reject H₀

One-tailed (left):
If test statistic < critical value → Reject H₀

Example:

Coin flip test: z = 3.0
Critical value (α = 0.05, two-tailed): ±1.96

Decision:
|3.0| > 1.96 → Reject H₀

This matches our p-value decision (p = 0.0027 < 0.05)

📏 Test Statistic - “The Evidence Score”

The test statistic is a single number that summarizes your evidence:

• Could be a Z-score, t-score, or other value
• Tells you “how far” your sample is from what’s claimed
• Bigger numbers (positive or negative) = more unusual evidence

Formula Pattern:

Test Statistic = (What you observed - What was claimed) / Standard Error

3. The 5-Step Process

Every hypothesis test follows these steps:

Step 1: State Your Hypotheses 📝

Write down H₀ and H₁ clearly.

Example:

H₀: The coin is fair (p = 0.50)
H₁: The coin is NOT fair (p ≠ 0.50)
α = 0.05

Step 2: Collect Your Data 📊

Gather your sample and calculate statistics.

Example:

Flipped coin 100 times
Got 65 heads
Sample proportion = 65/100 = 0.65

Step 3: Calculate the Test Statistic 🧮

Use the appropriate formula for your test.

Example:

Z = (0.65 - 0.50) / √(0.50 × 0.50 / 100)
Z = 0.15 / 0.05
Z = 3.0

Step 4: Find the P-Value 🔍

Look up or calculate the p-value.

Example:

For Z = 3.0 (two-tailed)
p-value ≈ 0.003

Step 5: Make Your Decision ⚖️

Compare p-value to α and conclude.

Example:

p-value (0.003) < α (0.05)
→ REJECT H₀
→ The coin is NOT fair!

4. Types of Errors

Even with good evidence, we can make mistakes!

🚨 Type I Error - “False Alarm”

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

• Like convicting an innocent person
• Probability = α (significance level)

Example:

• H₀: “The coin is fair” (and it really IS fair)
• But you reject it anyway because you got unlucky with your sample
• You cry “BIASED!” when it’s actually fair

🚨 Type II Error - “Missed Detection”

What it is: Failing to reject H₀ when it’s actually FALSE

• Like letting a guilty person go free
• Probability = β (beta)

Example:

• H₀: “The coin is fair” (but it’s actually BIASED)
• But you fail to reject it because your sample wasn’t extreme enough
• You say “It’s fair!” when it’s actually biased

The Trade-Off Table

Memory Trick:

• Type I: “I thought I saw something, but I was wrong” (false positive)
• Type II: “I missed it!” (false negative)

5. One-Tailed vs Two-Tailed Tests

🎯 Two-Tailed Test - “Is it Different?”

You care if the value is different in EITHER direction (higher OR lower).

H₁ uses: ≠ (not equal to)

Example:

H₀: Average height = 170 cm
H₁: Average height ≠ 170 cm
(Could be taller OR shorter)

Visual:

        Reject H₀     Don't Reject H₀     Reject H₀
        (2.5%)           (95%)              (2.5%)
    |---------|========================|---------|
   -∞      -1.96         0            1.96      +∞

🎯 One-Tailed Test (Right) - “Is it Greater?”

You only care if the value is HIGHER.

H₁ uses: > (greater than)

Example:

H₀: Average height = 170 cm
H₁: Average height > 170 cm
(Only care if TALLER)

Visual:

        Don't Reject H₀              Reject H₀
             (95%)                      (5%)
    |========================|---------|
   -∞                      1.645      +∞

🎯 One-Tailed Test (Left) - “Is it Less?”

You only care if the value is LOWER.

H₁ uses: < (less than)

Example:

H₀: Average height = 170 cm
H₁: Average height < 170 cm
(Only care if SHORTER)

Visual:

        Reject H₀         Don't Reject H₀
          (5%)                 (95%)
    |---------|========================|
   -∞      -1.645                     +∞

When to Use Which?

Use Two-Tailed when:

• You care about differences in BOTH directions
• You’re asking “Is it different?”
• Most common in practice

Use One-Tailed when:

• You only care about ONE direction
• You’re asking “Is it greater?” or “Is it less?”
• You have a specific directional hypothesis

6. Complete Manual Calculation Walkthrough

Let’s work through a complete example by hand to see exactly how everything is calculated!

📝 Problem Setup

Scenario: A coffee shop claims their average wait time is 5 minutes. You visit 36 times and record wait times. Your sample has:

• Sample mean (x̄) = 5.8 minutes
• Sample standard deviation (s) = 2.4 minutes
• Sample size (n) = 36

Question: Is the coffee shop’s claim accurate? Use α = 0.05.

Step 1: State Hypotheses

H₀: μ = 5 minutes (The claim is true)
H₁: μ ≠ 5 minutes (The claim is false)
α = 0.05
Test type: Two-tailed (we care if it's different in either direction)

Step 2: Choose the Test

Decision: Use one-sample t-test because:

• ✅ We’re comparing sample mean to a claimed value
• ✅ Population standard deviation (σ) is unknown
• ✅ We only have sample standard deviation (s)

Step 3: Calculate Test Statistic

Formula:

t = (x̄ - μ₀) / (s / √n)

Calculation:

Step 3a: Calculate standard error (SE)
SE = s / √n
SE = 2.4 / √36
SE = 2.4 / 6
SE = 0.4 minutes

Step 3b: Calculate t-statistic
t = (x̄ - μ₀) / SE
t = (5.8 - 5.0) / 0.4
t = 0.8 / 0.4
t = 2.0

Interpretation: Our sample mean is 2.0 standard errors above the claimed mean.

Step 4: Find Degrees of Freedom

df = n - 1
df = 36 - 1
df = 35

Step 5: Find P-Value

Method A: Using T-Table

1. Look up t = 2.0 with df = 35 in t-table
2. Find the area in one tail
3. For t = 2.0, df = 35: one-tail area ≈ 0.027
4. Two-tailed p-value = 2 × 0.027 = 0.054

Method B: Using Python (more accurate)

from scipy import stats
t_stat = 2.0
df = 35
p_value = 2 * (1 - stats.t.cdf(abs(t_stat), df))
print(f"p-value = {p_value:.4f}")
# Output: p-value = 0.0530

Step 6: Find Critical Value (Alternative Method)

Instead of p-value, we can use critical values:

For α = 0.05, two-tailed, df = 35:
Critical t-value = ±2.030 (from t-table)

Decision rule:
If |t| > 2.030 → Reject H₀
If |t| ≤ 2.030 → Fail to reject H₀

Our test statistic: |2.0| = 2.0
Comparison: 2.0 < 2.030
Decision: Fail to reject H₀

Step 7: Make Decision

Using p-value approach:

p-value = 0.0530
α = 0.05

Since 0.0530 > 0.05:
→ Fail to reject H₀

Using critical value approach:

|t| = 2.0
Critical value = 2.030

Since 2.0 < 2.030:
→ Fail to reject H₀

Both methods agree!

Step 8: State Conclusion

Statistical conclusion: “At the 0.05 significance level, we do not have sufficient evidence to reject the claim that the average wait time is 5 minutes.”

Practical conclusion: “While our sample average (5.8 minutes) is higher than the claimed 5 minutes, this difference could reasonably occur by chance. The coffee shop’s claim cannot be rejected based on this data.”

Note: The p-value (0.053) is very close to α (0.05)! This is a borderline case. With a slightly larger sample or slightly higher sample mean, we might reject H₀.

📊 Visual Summary of Calculations

Given Data:
├─ x̄ = 5.8 minutes
├─ μ₀ = 5.0 minutes
├─ s = 2.4 minutes
├─ n = 36
└─ α = 0.05

Calculations:
├─ SE = s/√n = 2.4/6 = 0.4
├─ t = (x̄-μ₀)/SE = 0.8/0.4 = 2.0
├─ df = n-1 = 35
├─ p-value = 0.053
└─ Critical value = ±2.030

Decision:
├─ p-value (0.053) > α (0.05) ✓
├─ |t| (2.0) < critical (2.030) ✓
└─ Conclusion: Fail to reject H₀

7. Complete Python Example

Let’s put it all together with a real example!

Example: Testing a Coin for Fairness

The Story: You suspect a coin might be biased. You flip it 100 times and get 65 heads. Is the coin fair?

# Step 0: Import tools
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

print("🪙 COIN FAIRNESS TEST - COMPLETE WALKTHROUGH")
print("="*70)

# ============================================================
# STEP 1: STATE THE HYPOTHESES
# ============================================================
print("\n📝 STEP 1: STATE THE HYPOTHESES")
print("-"*70)
print("H₀ (Null): The coin is fair (p = 0.50)")
print("H₁ (Alternative): The coin is NOT fair (p ≠ 0.50)")
print("Significance level: α = 0.05")
print("Test type: Two-tailed (we care if it's biased in either direction)")

# ============================================================
# STEP 2: COLLECT THE DATA
# ============================================================
print("\n📊 STEP 2: COLLECT THE DATA")
print("-"*70)

n = 100                    # Number of flips
heads = 65                 # Number of heads observed
p_observed = heads / n     # Sample proportion
p_claimed = 0.50           # Claimed proportion (fair coin)

print(f"Number of flips: {n}")
print(f"Heads observed: {heads}")
print(f"Sample proportion: {p_observed:.2f} ({p_observed*100:.0f}%)")
print(f"Claimed proportion: {p_claimed:.2f} ({p_claimed*100:.0f}%)")
print(f"Difference: {(p_observed - p_claimed)*100:.0f} percentage points")

# ============================================================
# STEP 3: CALCULATE THE TEST STATISTIC
# ============================================================
print("\n🧮 STEP 3: CALCULATE THE TEST STATISTIC")
print("-"*70)

# Standard error for proportion
se = np.sqrt(p_claimed * (1 - p_claimed) / n)
print(f"Standard Error: {se:.4f}")

# Z-statistic
z_stat = (p_observed - p_claimed) / se
print(f"Z-statistic: {z_stat:.4f}")

print("\n💡 What does this mean?")
print(f"   Our sample proportion is {z_stat:.2f} standard errors")
print(f"   away from the claimed proportion.")
print(f"   That's {'pretty far' if abs(z_stat) > 2 else 'not too far'}!")

# ============================================================
# STEP 4: FIND THE P-VALUE
# ============================================================
print("\n🔍 STEP 4: FIND THE P-VALUE")
print("-"*70)

# Two-tailed p-value
p_value = 2 * (1 - stats.norm.cdf(abs(z_stat)))
print(f"P-value: {p_value:.4f}")

print("\n💡 What does this mean?")
print(f"   If the coin were really fair, there's only a")
print(f"   {p_value*100:.2f}% chance of getting results this extreme")
print(f"   or more extreme just by random luck.")

# ============================================================
# STEP 5: MAKE YOUR DECISION
# ============================================================
print("\n⚖️  STEP 5: MAKE YOUR DECISION")
print("-"*70)

alpha = 0.05
print(f"Significance level (α): {alpha}")
print(f"P-value: {p_value:.4f}")
print(f"Comparison: p-value {'<' if p_value < alpha else '>'} α")

print("\n" + "="*70)
print("🎯 FINAL DECISION:")
print("="*70)

if p_value < alpha:
    print("✅ REJECT H₀")
    print("\n📊 Conclusion:")
    print("   There IS sufficient evidence to conclude that the coin is biased.")
    print(f"   The probability of getting {heads} or more heads (or {100-heads} or fewer)")
    print(f"   in {n} flips of a fair coin is only {p_value*100:.2f}%.")
    print("   This is too unlikely to be just random chance!")
else:
    print("❌ FAIL TO REJECT H₀")
    print("\n📊 Conclusion:")
    print("   There is NOT sufficient evidence to conclude that the coin is biased.")
    print(f"   Getting {heads} heads in {n} flips could reasonably happen")
    print("   with a fair coin just by random chance.")

print("="*70)

# ============================================================
# BONUS: VISUALIZE THE RESULTS
# ============================================================
print("\n📈 Creating visualizations...")

fig, axes = plt.subplots(1, 3, figsize=(16, 5))

# Plot 1: Pie chart of results
axes[0].pie([heads, n-heads], labels=['Heads', 'Tails'], 
            autopct='%1.0f%%', colors=['gold', 'silver'], startangle=90)
axes[0].set_title(f'Coin Flip Results\n({n} flips)', fontweight='bold')

# Plot 2: Comparison to expected
categories = ['Expected\n(Fair Coin)', 'Observed']
proportions = [p_claimed, p_observed]
colors = ['blue', 'red' if p_value < alpha else 'orange']

bars = axes[1].bar(categories, proportions, color=colors, alpha=0.7, edgecolor='black')
axes[1].axhline(p_claimed, color='blue', linestyle='--', linewidth=2, label='Fair coin (50%)')
axes[1].set_ylabel('Proportion of Heads')
axes[1].set_title('Expected vs Observed', fontweight='bold')
axes[1].set_ylim(0, 1)
axes[1].legend()
axes[1].grid(True, alpha=0.3, axis='y')

# Add percentage labels on bars
for bar, prop in zip(bars, proportions):
    height = bar.get_height()
    axes[1].text(bar.get_x() + bar.get_width()/2., height + 0.02,
                f'{prop*100:.0f}%', ha='center', va='bottom', fontweight='bold')

# Plot 3: Z-distribution with test statistic
x = np.linspace(-4, 4, 1000)
y = stats.norm.pdf(x)

axes[2].plot(x, y, 'b-', linewidth=2, label='Standard Normal Distribution')
axes[2].axvline(z_stat, color='red', linestyle='--', linewidth=2, 
                label=f'Our Z-score = {z_stat:.2f}')

# Critical values for two-tailed test
z_critical = stats.norm.ppf(1 - alpha/2)
axes[2].axvline(z_critical, color='green', linestyle='--', linewidth=1.5, 
                label=f'Critical values = ±{z_critical:.2f}')
axes[2].axvline(-z_critical, color='green', linestyle='--', linewidth=1.5)

# Shade rejection regions
x_left = np.linspace(-4, -z_critical, 100)
x_right = np.linspace(z_critical, 4, 100)
axes[2].fill_between(x_left, stats.norm.pdf(x_left), alpha=0.3, color='red', 
                     label='Rejection Regions')
axes[2].fill_between(x_right, stats.norm.pdf(x_right), alpha=0.3, color='red')

axes[2].set_xlabel('Z-score')
axes[2].set_ylabel('Probability Density')
axes[2].set_title('Hypothesis Test Visualization', fontweight='bold')
axes[2].legend()
axes[2].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

print("\n✅ Analysis complete!")

Expected Output:

🪙 COIN FAIRNESS TEST - COMPLETE WALKTHROUGH
======================================================================

📝 STEP 1: STATE THE HYPOTHESES
----------------------------------------------------------------------
H₀ (Null): The coin is fair (p = 0.50)
H₁ (Alternative): The coin is NOT fair (p ≠ 0.50)
Significance level: α = 0.05
Test type: Two-tailed (we care if it's biased in either direction)

📊 STEP 2: COLLECT THE DATA
----------------------------------------------------------------------
Number of flips: 100
Heads observed: 65
Sample proportion: 0.65 (65%)
Claimed proportion: 0.50 (50%)
Difference: 15 percentage points

🧮 STEP 3: CALCULATE THE TEST STATISTIC
----------------------------------------------------------------------
Standard Error: 0.0500
Z-statistic: 3.0000

💡 What does this mean?
   Our sample proportion is 3.00 standard errors
   away from the claimed proportion.
   That's pretty far!

🔍 STEP 4: FIND THE P-VALUE
----------------------------------------------------------------------
P-value: 0.0027

💡 What does this mean?
   If the coin were really fair, there's only a
   0.27% chance of getting results this extreme
   or more extreme just by random luck.

⚖️  STEP 5: MAKE YOUR DECISION
----------------------------------------------------------------------
Significance level (α): 0.05
P-value: 0.0027
Comparison: p-value < α

======================================================================
🎯 FINAL DECISION:
======================================================================
✅ REJECT H₀

📊 Conclusion:
   There IS sufficient evidence to conclude that the coin is biased.
   The probability of getting 65 or more heads (or 35 or fewer)
   in 100 flips of a fair coin is only 0.27%.
   This is too unlikely to be just random chance!
======================================================================

🎓 Summary - What You Learned

Key Concepts

1. Hypothesis Testing = Making decisions about populations using sample data
2. H₀ (Null) = The boring claim we try to disprove
3. H₁ (Alternative) = The interesting claim we try to prove
4. P-value = “How weird is our evidence?” (< 0.05 = very weird!)
5. α (Alpha) = Our threshold for “weird enough” (usually 0.05)
6. Test Statistic = A score measuring how far our sample is from the claim

The 5-Step Process

1. State hypotheses (H₀ and H₁)
2. Collect data
3. Calculate test statistic
4. Find p-value
5. Make decision (reject or don't reject H₀)

Decision Rule

If p-value < α:  REJECT H₀ (found something interesting!)
If p-value ≥ α:  DON'T REJECT H₀ (not enough evidence)

Types of Errors

• Type I Error: False alarm (reject H₀ when it’s true)
• Type II Error: Missed detection (don’t reject H₀ when it’s false)

Test Types

• Two-tailed: H₁ uses ≠ (different in either direction)
• One-tailed (right): H₁ uses > (greater than)
• One-tailed (left): H₁ uses < (less than)

🎯 What’s Next?

Now that you understand the basics of hypothesis testing, you’re ready to learn about specific tests!

Continue to:

• Chapter 4.1: Single Sample Tests (Z-test, T-test)
• Chapter 4.2: Two Sample Tests (Independent, Paired)
• Chapter 4.3: Proportion Tests

Each chapter will teach you WHEN to use each test and HOW to do it in Python!

💡 Practice Problems

Problem 1: Quick Decision Practice

For each scenario, decide: Reject H₀ or Don’t Reject H₀?

a) p-value = 0.03, α = 0.05 b) p-value = 0.08, α = 0.05 c) p-value = 0.001, α = 0.01 d) p-value = 0.06, α = 0.10

Problem 2: Identify the Hypotheses

A company claims their light bulbs last 1000 hours on average. You think they last less. Write H₀ and H₁.

Problem 3: Error Types

You test a new drug. The drug actually works (H₀ is false), but your test fails to reject H₀. What type of error is this?

Remember: Hypothesis testing is about making informed decisions with data. You’re not looking for absolute proof - you’re weighing evidence and making the best decision you can with the information available! 🎯

💻 Hands-On Practice

Ready to practice hypothesis testing with real data?

Check out the Jupyter Notebook with Hypothesis Testing Examples using the Tips dataset!

What’s included:

• ✅ Complete 5-step process examples
• ✅ Real dataset (restaurant tips)
• ✅ One-sample t-test (Is average tip $3.00?)
• ✅ Two-sample t-test (Do smokers tip differently?)
• ✅ Proportion test (Weekend vs weekday dining)
• ✅ p-value visualizations
• ✅ Decision-making examples

To run the notebook:

1. Install: pip install pandas numpy matplotlib seaborn scipy statsmodels jupyter
2. Download the notebook from the repository
3. Run: jupyter notebook chapter3_hypothesis_testing_examples.ipynb

📖 See the notebooks README for full instructions!

Ready to learn specific tests? Head to Chapter 4! 🚀