HOW TO SOLVE HYPOTHESIS TESTING: COMPLETE STEP-BY-STEP STATS GUIDE

Introduction

Hypothesis testing intimidates most students because it feels abstract. You’re asked to “test” something, but the logic remains fuzzy. The truth: hypothesis testing is a structured decision-making process that becomes intuitive once you follow the five-step framework. This guide walks you through each step with real worked examples, software implementations, and common mistakes to avoid.

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The 5-Step Hypothesis Testing Framework

All hypothesis tests follow the same logical structure, regardless of test type. Master this framework and you can apply it to any test.

5-Step Hypothesis Testing Framework: From Hypothesis to Decision 

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Step 1: State the Hypotheses (H₀ and H₁)

Every hypothesis test begins with two competing claims about the population parameter.

Null Hypothesis (H₀): The default assumption—usually “no effect” or “no difference.”

• Example: μ = 100 (the population mean equals 100)
• Example: p₁ = p₂ (the two populations have equal proportions)
• Always contains “=” (equality)

Alternative Hypothesis (H₁): Your research claim—what you want to prove.

• Two-tailed: H₁: μ ≠ 100 (different, either direction)
• One-tailed (left): H₁: μ < 100 (less than)
• One-tailed (right): H₁: μ > 100 (greater than)

Key Decision: Should you state your claim as H₀ or H₁?

Best practice: State your claim as H₁ (the alternative). Here’s why: If evidence supports H₁, you have a stronger result (“We found evidence for…”) than if you fail to reject H₀ (“We didn’t find evidence against…”).

Common Mistake: Choosing between hypotheses after seeing the data. This is “cart before the horse” and invalidates your test. Hypotheses must be determined beforehand.youtube​

Step 2: Choose Significance Level (α)

The significance level (alpha) is your Type I error tolerance—the probability of falsely rejecting a true null hypothesis.

Standard: α = 0.05 (5% false positive rate tolerated)
Conservative: α = 0.01 (1% false positive rate, stricter)
Lenient: α = 0.10 (10%, less common)

In Plain Language: If α = 0.05, you’re willing to be wrong 5% of the time by claiming an effect exists when it doesn’t.

When to adjust α:

• Medical/safety testing: Use α = 0.01 (lower tolerance for false positives)
• Exploratory research: Can use α = 0.10
• Standard: α = 0.05

Step 3: Select Test Statistic

The test statistic is a single number calculated from your sample data that summarizes evidence against H₀. Different data types require different tests.

When to use each test:

Data Type	Test	Formula	Example
1 mean vs. population	One-sample t-test	t = (x̄ – μ₀) / (s/√n)	Is average engineer height 5’10”?
2 independent means	Two-sample t-test	t = (x̄₁ – x̄₂) / √(s₁²/n₁ + s₂²/n₂)	Do men and women earn differently?
2 paired means	Paired t-test	t = (d̄) / (sd/√n)	Does weight change before/after diet?
2+ categorical variables	Chi-square test	χ² = Σ(O – E)² / E	Is product preference independent of gender?
1 proportion vs. population	One-sample z-test	z = (p̂ – p₀) / √(p₀(1-p₀)/n)	Is defect rate 5%?

How to choose:

• Comparing means of continuous data → t-test or z-test
• Comparing counts or proportions → chi-square test
• Small sample (n < 30) → t-test
• Large sample (n ≥ 30) → can use z-test (or t-test still valid)

Step 4: Calculate Test Statistic & Find P-value

This is mechanical computation—most done by software. The test statistic measures how far your sample result is from the null hypothesis value, expressed in standard errors.

P-value interpretation:
The p-value is the probability of observing your sample result (or more extreme) if H₀ were true.

• Small p-value (< 0.05): Unlikely result under H₀ → suggests H₀ is false
• Large p-value (≥ 0.05): Likely result under H₀ → H₀ is plausible

Not the probability that H₀ is true. This is the most common misconception.yourstatsguru​

Step 5: Make Decision & Interpret Results

Decision rule:

• If p-value < α: Reject H₀ (statistically significant result)
• If p-value ≥ α: Fail to reject H₀ (not statistically significant)

Interpretation language matters:

✓ Correct: “We reject H₀ and conclude there is significant evidence for H₁.”
✗ Incorrect: “We proved H₁” or “H₀ is false”

✓ Correct: “We failed to reject H₀; insufficient evidence for H₁.”
✗ Incorrect: “H₀ is true” or “No effect exists”

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Worked Example 1: One-Sample T-Test (Engineering Context)

Scenario: A metal rod manufacturer claims rods average 100 mm in length. Quality control tests a random sample of 25 rods to verify this claim.

Data:

• Sample mean: x̄ = 101.5 mm
• Sample std. dev: s = 2.3 mm
• Sample size: n = 25
• Claimed population mean: μ₀ = 100 mm

Step 1: State Hypotheses

• H₀: μ = 100 (rods average 100 mm)
• H₁: μ ≠ 100 (rods differ from 100 mm, two-tailed)

Step 2: Choose α = 0.05

Step 3: Select One-Sample T-Test

Step 4: Calculate Test Statistic

t = (x̄ – μ₀) / (s / √n)
t = (101.5 – 100) / (2.3 / √25)
t = 1.5 / (2.3 / 5)
t = 1.5 / 0.46
t = 3.26

Degrees of freedom: df = n – 1 = 24

Find p-value: Using t-distribution table or software with t = 3.26, df = 24, two-tailed:
p-value ≈ 0.0038

Step 5: Make Decision

p-value (0.0038) < α (0.05) → Reject H₀

Interpretation:
“The sample provides strong evidence that rods differ significantly from the claimed 100 mm average (t(24) = 3.26, p = 0.0038). Quality control should investigate the manufacturing process.”

Worked Example 2: Two-Sample T-Test (A/B Testing Context)

Scenario: An e-commerce company tests two website designs to see if Design B increases average order value. They randomly assign customers to Design A (current) or Design B (test) and track average orders.

Data:

• Design A: n₁ = 150, x̄₁ = $52.40, s₁ = $18.20
• Design B: n₂ = 150, x̄₂ = $58.75, s₂ = $19.80

Step 1: State Hypotheses

• H₀: μ₁ = μ₂ (no difference in order value between designs)
• H₁: μ₁ ≠ μ₂ (designs differ in order value, two-tailed)

Step 2: Choose α = 0.05

Step 3: Select Two-Sample T-Test

Step 4: Calculate Test Statistic

t = (x̄₁ – x̄₂) / √(s₁²/n₁ + s₂²/n₂)
t = (52.40 – 58.75) / √((18.20²/150) + (19.80²/150))
t = -6.35 / √(2.205 + 2.613)
t = -6.35 / √4.818
t = -6.35 / 2.195
t = -2.89

Degrees of freedom: Approximation: df ≈ 298

Find p-value: Using t-distribution with t = -2.89, df ≈ 298, two-tailed:
p-value ≈ 0.0041

Step 5: Make Decision

p-value (0.0041) < α (0.05) → Reject H₀

Interpretation:
“Design B significantly increases average order value by $6.35 compared to Design A (t(298) = -2.89, p = 0.0041). This provides strong evidence to recommend rolling out Design B.”

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Worked Example 3: Chi-Square Test (Categorical Data)

Scenario: A university wants to know if student satisfaction with campus facilities differs by class year (freshmen, sophomores, juniors, seniors).

Survey Results:

Class	Satisfied	Unsatisfied	Total
Freshman	45	55	100
Sophomore	52	48	100
Junior	58	42	100
Senior	62	38	100
Total	217	183	400

Step 1: State Hypotheses

• H₀: Class year and satisfaction are independent (no association)
• H₁: Class year and satisfaction are associated (not independent)

Step 2: Choose α = 0.05

Step 3: Select Chi-Square Test of Independence

Step 4: Calculate Expected Frequencies & χ² Statistic

Expected frequency formula: E = (Row Total × Column Total) / N

For Freshman-Satisfied: E = (100 × 217) / 400 = 54.25
For Freshman-Unsatisfied: E = (100 × 183) / 400 = 45.75

(Continuing for all cells…)

Chi-square statistic:
χ² = Σ (Observed – Expected)² / Expected
χ² = (45-54.25)²/54.25 + (55-45.75)²/45.75 + … = 8.47

Degrees of freedom: df = (rows – 1) × (columns – 1) = (4-1) × (2-1) = 3

Find p-value: Using chi-square distribution with χ² = 8.47, df = 3:
p-value ≈ 0.037

Step 5: Make Decision

p-value (0.037) < α (0.05) → Reject H₀

Interpretation:
“There is significant association between class year and satisfaction with campus facilities (χ²(3) = 8.47, p = 0.037). Upper-class students report higher satisfaction than freshmen.”

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Common Mistakes & How to Avoid Them

Mistake 1: Setting Hypotheses After Seeing Datayoutube​

What students do: Calculate sample statistics, then write hypotheses based on results.

Why it’s wrong: This defeats hypothesis testing. You already know the answer from summary statistics.

How to fix it: Write hypotheses BEFORE data analysis. Hypothesis testing is a “blind guess” that you then test with data.

Mistake 2: Misinterpreting P-Valuesyourstatsguru​

Wrong: “The p-value is the probability H₀ is true” (0.05 = 5% chance H₀ true)

Correct: “The p-value is the probability of observing this data (or more extreme) if H₀ were true”

Example: p = 0.03 means “there’s a 3% chance we’d see results this extreme if H₀ were true”—NOT “3% chance H₀ is true.”

Mistake 3: Confusing Test Selection

Wrong: Using z-test for small samples (n < 30)

Correct: Use t-test for small samples; z-test for large samples or known population σ.

Wrong: Using t-test for categorical data (proportions)

Correct: Use chi-square for categorical; z-test or binomial for single proportion.

Mistake 4: Ignoring Type I & II Errorsbyjus+1​

Type I Error (False Positive): Rejecting H₀ when it’s actually true.

• Probability = α (your chosen significance level)
• Example: Concluding a drug works when it doesn’t

Type II Error (False Negative): Failing to reject H₀ when it’s actually false.

• Probability = β
• Example: Concluding a drug doesn’t work when it does

Key insight: You can’t minimize both errors simultaneously. Lowering α increases β. Choose based on consequences.

Mistake 5: Saying “Prove” or “Accept” H₀

Wrong: “We proved H₁” or “We accept H₀”

Correct: “We reject H₀ in favor of H₁” or “We fail to reject H₀”

Hypothesis testing provides evidence, not proof.

Software Walkthroughs

Excel: One-Sample T-Test

text

Data in cells A2:A26 (25 rod lengths)

 

Formula:

=T.TEST(A2:A26, 100, 2, 1)

 

Where:

– A2:A26 = data range

– 100 = hypothesized mean

– 2 = two-tailed test

– 1 = one-sample test

 

Result: p-value directly displayed

R: Two-Sample T-Test

r

# Create data

design_a <- rnorm(150, mean = 52.4, sd = 18.2)

design_b <- rnorm(150, mean = 58.75, sd = 19.8)

 

# Perform two-sample t-test

result <- t.test(design_a, design_b)

 

# View results

print(result)

# Shows: t-statistic, df, p-value, confidence interval

Python: Chi-Square Test

python

from scipy.stats import chi2_contingency

import pandas as pd

 

# Create contingency table

data = np.array([[45, 55], [52, 48], [58, 42], [62, 38]])

 

# Perform chi-square test

chi2, p_value, dof, expected = chi2_contingency(data)

 

print(f”Chi-square statistic: {chi2:.2f}”)

print(f”P-value: {p_value:.4f}”)

print(f”Degrees of freedom: {dof}”)

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Practice Problems

Problem 1: 

A coffee shop claims its espresso shots average 30 mL. A customer measures 12 shots: mean = 28.5 mL, SD = 1.8 mL. Test at α = 0.05.

• Solution: t(11) = -2.88, p ≈ 0.015. Reject H₀. Shots are significantly smaller than claimed.

Problem 2: 

Two teaching methods are tested. Method A: n = 40, mean = 75, SD = 12. Method B: n = 40, mean = 79, SD = 13. Test at α = 0.05.

• Solution: t ≈ -1.35, p ≈ 0.18. Fail to reject H₀. No significant difference.

Problem 3:

 Survey data: Does preference for Product X differ by age group?

• Younger: 70 prefer, 30 don’t. Older: 50 prefer, 50 don’t. Test at α = 0.05.
• Solution: χ² ≈ 8.0, p ≈ 0.005. Reject H₀. Strong association between age and preference.

Key Takeaways

1. Follow the 5-step framework: Hypotheses → α → Test Selection → Calculation → Decision
2. P-value is NOT probability H₀ is true—it’s probability of data given H₀
3. Choose test based on data type: means → t-test; counts → chi-square
4. Small p-value < α means reject H₀—statistically significant
5. State hypotheses before seeing data—avoid “cart before horse”
6. Type I error (false positive) = α; Type II error (false negative) = β
7. Use software for calculations—focus on interpretation

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