Hypothesis Testing Guide

Introduction

Learning to write proper hypotheses is a fundamental skill in statistics. This guide will walk you through the process with clear steps, detailed explanations, and important reminders. Take your time with each section to build a solid foundation.

What Is Hypothesis Testing?

The Basic Idea:

Hypothesis testing is a method for making decisions about a population based on a sample. Think of it like a courtroom trial: we start by assuming innocence (the null hypothesis) and only reject that assumption if we have strong evidence against it.

What Hypothesis Testing Actually Tests:

When you conduct a hypothesis test, you’re asking: “If the null hypothesis were true, how likely would I be to get data like what I observed?” If your data would be very unlikely under the null hypothesis, you have evidence to reject it.

Important limitation:

Hypothesis testing tells you whether your data is consistent or inconsistent with a particular claim about a population parameter. It does NOT tell you:

Why the parameter has a certain value
What causes the parameter to be what it is
How one variable affects another variable
Whether two variables are related to each other

Think of it this way: A thermometer can tell you the temperature is 75°F, but it cannot tell you why it’s 75°F or what caused it to reach that temperature. Similarly, basic hypothesis testing describes characteristics of a population but doesn’t explain causes or relationships.

Step 1: Identify Your Research Question

Start by clearly stating what you want to investigate. Your question should be specific and focused on a single measurable characteristic of one group.

Good research questions for beginners:

What is the average height of students at my university?
What proportion of registered voters in my city support the new park proposal?
Is the average amount of sleep college students get equal to 7 hours per night?

Why these are good: Each question focuses on describing ONE characteristic (height, proportion, sleep duration) of ONE defined group (university students, city voters, college students).

Questions to avoid as a beginner:

Do students who study more get better grades? (This implies a relationship between two variables)
Does drinking coffee improve test performance? (This implies causation)
Are men taller than women? (This compares two groups, which requires different tests)

Practice tip: If your question contains words like “causes,” “affects,” “related to,” “compared to,” or “correlation,” you’re likely asking about relationships or causation, which goes beyond basic hypothesis testing.

Step 2: Identify the Population Parameter

A parameter is a numerical characteristic of an entire population. You need to identify which parameter you’re studying.

Common parameters for beginners:

μ (mu) - Population Mean:
This is the average value for an entire population. Use this when your data is numerical and you’re interested in the typical or average amount.

Example: The average number of hours students study per week
Example: The mean temperature in July
Example: The average price of textbooks

p - Population Proportion:
This is the percentage or fraction of a population that has a certain characteristic. Use this when your data is categorical (yes/no, success/failure, prefer A or B).

Example: The proportion of students who own a car
Example: The percentage of voters who support a candidate
Example: The fraction of products that are defective

How to choose: Ask yourself: “Am I calculating an average of numbers (mean) or counting how many have a certain characteristic (proportion)?”

Why we focus on one parameter: Basic hypothesis testing examines one parameter at a time. This keeps things simple and clear. You’re describing what IS, not comparing or explaining.

Step 3: Check That Your Data Is Obtainable

Before writing hypotheses, make sure you can realistically collect the data you need.

Ask yourself:

Can I survey or measure a sample from this population?
Is the information publicly available or can I access it?
Do I have the time and resources needed?
Is it ethical and legal to collect this data?
Can the characteristic be measured objectively?

Realistic examples:

Surveying students in your classes about their study habits
Recording the prices of items at local stores
Measuring the height of plants you’re growing
Counting how many of your friends prefer option A vs. option B

Unrealistic examples for students:

Accessing private medical records of hospital patients
Surveying all residents of an entire state
Data that requires expensive laboratory equipment
Information that people are unlikely to answer honestly

Why this matters: If you can’t collect the data, you can’t test your hypothesis. Always think about feasibility before committing to a research question.

Step 4: Write the Null Hypothesis (H₀)

The null hypothesis is your starting assumption. It states that a population parameter equals a specific value.

The format is always: H₀: [parameter] = [specific number]

Key points about the null hypothesis:

It uses “equals” only: The null hypothesis ALWAYS uses an equals sign (=). Never use ≠, <, or > in a null hypothesis.
It represents the status quo: This is what you assume is true unless your data proves otherwise. Think of it as the “nothing special is happening” statement.
It must be specific: You need an exact number, not a range or vague statement.

Examples with explanations:

Example 1: H₀: μ = 30
- Translation: “The population mean equals 30”
- What this means: We’re assuming the true average in the entire population is exactly 30 (units depend on context: 30 minutes, 30 inches, 30 dollars, etc.)

Example 2: H₀: p = 0.50
- Translation: “The population proportion equals 0.50” or “50%”
- What this means: We’re assuming that exactly half (50%) of the population has the characteristic we’re studying

Example 3: H₀: μ = 7
- Translation: “The population mean equals 7”
- Context: If studying sleep hours, this means we assume the average person sleeps exactly 7 hours

Where does the specific number come from?

Previous research or historical data
A claimed value (like a manufacturer’s claim)
A theoretical expectation
A value representing “no effect” or “no preference” (like 0.50 for a proportion when testing if something is 50-50)

Step 5: Write the Alternative Hypothesis (H₁ or Hₐ)

The alternative hypothesis states what you’re actually testing for. This is what you suspect might be true instead of the null hypothesis.

Two-tailed tests (most common for beginners):

Format: H₁: [parameter] ≠ [specific number]

The ≠ symbol means “not equal to” or “different from.” This tests whether the parameter is either higher OR lower than the null value.

Example: H₁: μ ≠ 30
- Translation: “The population mean is not equal to 30”
- What you’re testing: Is the true average different from 30 in either direction?
- When to use: When you want to detect ANY difference, whether it’s above or below the null value

Why two-tailed tests are beginner-friendly: They’re more conservative and flexible. You don’t have to predict which direction the difference will be.

One-tailed tests (use only when you have a specific direction in mind):

Greater than: H₁: μ > 30
- Translation: “The population mean is greater than 30”
- When to use: Only when you specifically want to test if the value is HIGHER than the null value
- Example scenario: A company claims their battery lasts “more than 10 hours”

Less than: H₁: μ < 30
- Translation: “The population mean is less than 30”
- When to use: Only when you specifically want to test if the value is LOWER than the null value
- Example scenario: A new drug claims to reduce cholesterol “below 200”

Important decision: When in doubt, use a two-tailed test (≠). Only use one-tailed tests when you have a strong reason to look in only one direction.

Step 6: Select the Appropriate Test Type

Different research questions require different statistical tests. Here’s how to match your hypotheses to the right test.

For testing a population mean (μ):

Use a one-sample t-test when:

Your parameter is μ (a mean/average)
You have numerical data (heights, times, prices, scores, etc.)
You want to test if the population mean equals a specific value
You’re examining ONE group only

Example: Testing if the average study time of college students equals 5 hours per week

H₀: μ = 5 hours
H₁: μ ≠ 5 hours
Test: One-sample t-test

For testing a population proportion (p):

Use a one-sample proportion test when:

Your parameter is p (a proportion/percentage)
Your data is categorical (yes/no, success/failure, category A or B)
You want to test if the population proportion equals a specific value
You’re examining ONE group only

Example: Testing if 60% of students prefer online classes

H₀: p = 0.60
H₁: p ≠ 0.60
Test: One-sample proportion test

Quick decision guide:

Calculating an average of numbers? → Use t-test for means
Counting what percentage has a trait? → Use proportion test

Understanding Significance Level (α)

After formulating your hypotheses, you need to choose a significance level before collecting data. This is a crucial part of hypothesis testing that beginners often find confusing.

What is a significance level?

The significance level, denoted by α (alpha), is the probability of rejecting the null hypothesis when it’s actually true. In simpler terms, it’s your tolerance for making a “false alarm” error.

Common significance levels:

α = 0.05 (most common) = 5% significance level
α = 0.01 (more strict) = 1% significance level
α = 0.10 (more lenient) = 10% significance level

What α = 0.05 means in plain English:

“I’m willing to accept a 5% chance of concluding the null hypothesis is false when it’s actually true.”

Or think of it this way: If you repeated your study 100 times and the null hypothesis was actually true every time, you’d expect to incorrectly reject it about 5 times just due to random chance.

How to choose your significance level:

Use α = 0.05 when:

You’re doing a standard classroom assignment or typical research
The consequences of a wrong conclusion are moderate
This is the default in most situations

Use α = 0.01 when:

You want to be more certain before rejecting the null hypothesis
The consequences of a false conclusion are serious
Example: Medical research where false conclusions could harm people

Use α = 0.10 when:

You’re doing exploratory research
You’re okay with less stringent evidence
You’re more worried about missing a real effect than making a false alarm

Important: You must choose your significance level BEFORE collecting data and running your test. You cannot change it after seeing your results, as this would be cheating.

What happens in a hypothesis test?

You calculate a p-value from your data (your statistical software or test does this)
You compare the p-value to your chosen α
Decision rules:
- If p-value ≤ α: Reject the null hypothesis (your data is unlikely if H₀ were true)
- If p-value > α: Fail to reject the null hypothesis (your data is reasonably consistent with H₀)

Example:

You chose α = 0.05
Your test produces p-value = 0.03
Since 0.03 < 0.05, you reject H₀
Interpretation: “At the 5% significance level, there is sufficient evidence to conclude the population mean is not equal to 30.”

What the significance level does NOT mean:

It is NOT the probability that the null hypothesis is true
It is NOT the probability that your conclusion is wrong
It does NOT tell you how important or large the effect is

Critical Rules to Remember

1. Avoid Implying Causation

Hypothesis testing describes what exists in a population but does NOT establish what causes it.

Why causation requires more than hypothesis testing:

To establish causation, you need:

Controlled experiments with random assignment
Comparison groups (treatment vs. control)
Ruling out confounding variables
Multiple studies showing consistent effects

A simple hypothesis test cannot do these things because it only describes one characteristic of one group at one point in time.

Wrong hypotheses (implying causation):

❌ “Studying causes test scores to increase”
❌ “Exercise causes weight loss”
❌ “Fertilizer causes plants to grow taller”

Right hypotheses (descriptive only):

✓ “The average test score is 85” (H₀: μ = 85)
✓ “The average weight is 150 pounds” (H₀: μ = 150)
✓ “The average plant height is 12 inches” (H₀: μ = 12)

Key insight: Notice that the correct hypotheses don’t mention studying, exercise, or fertilizer at all. They simply describe a characteristic of the group being measured.

2. Avoid Implying Relationships Between Variables

Basic hypothesis testing examines one variable at a time, not how two variables relate to each other.

Why relationship testing is different:

Testing relationships requires:

Measuring TWO variables for each subject
Correlation analysis or regression (more advanced techniques)
Different hypothesis structures
Different statistical tests entirely

Wrong hypotheses (implying relationships):

❌ “As study time increases, grades increase”
❌ “Taller people weigh more”
❌ “Age is related to income”

Right hypotheses (single variable):

✓ “The average study time is 5 hours” (H₀: μ = 5)
✓ “The average weight is 160 pounds” (H₀: μ = 160)
✓ “The average income is $50,000” (H₀: μ = 50,000)

Red flag words: If your research question includes words like “relationship,” “correlation,” “associated with,” “affects,” “depends on,” or “compared to,” you’re asking about relationships, which goes beyond basic hypothesis testing.

3. Use Proper Notation

Statistical notation has specific meanings and rules.

Population parameters (use these in hypotheses):

μ (mu) - population mean
p - population proportion
σ (sigma) - population standard deviation

Sample statistics (never use these in hypotheses):

x̄ (x-bar) - sample mean
p̂ (p-hat) - sample proportion
s - sample standard deviation

Why the distinction matters: Hypotheses are always about populations (the entire group you’re interested in), not samples (the subset you actually measure). You use samples to make inferences about populations, but your hypotheses must be stated in terms of population parameters.

Examples:

✓ Correct: H₀: μ = 100
❌ Wrong: H₀: x̄ = 100

4. Be Specific and Testable

Vague statements cannot be tested statistically.

Vague vs. Specific:

❌ Vague: “Students are tall”
- Problem: What does “tall” mean? What’s the standard?

✓ Specific: H₀: μ = 68 inches
- This is testable: collect height data and compare to 68 inches

❌ Vague: “Most people like the product”
- Problem: What does “most” mean exactly?

✓ Specific: H₀: p = 0.51
- This is testable: survey people and calculate the proportion who like it

Requirements for good hypotheses:

Uses exact numbers, not words like “high,” “low,” “most,” or “few”
Defines the population clearly
States the parameter precisely
Can be tested with data you can collect

Complete Examples with Full Explanations

Example 1: Testing a Mean

Scenario: A university claims that students spend an average of 15 hours per week studying. You want to test this claim.

Step-by-step:

Research question: Do students at this university spend an average of 15 hours per week studying?
Population: All students at the university
Variable: Hours spent studying per week (numerical data)
Parameter: μ (population mean study hours)
Null hypothesis: H₀: μ = 15 hours
- Interpretation: The true average study time for all students is 15 hours
Alternative hypothesis: H₁: μ ≠ 15 hours
- Interpretation: The true average is different from 15 hours (could be higher or lower)
- Type: Two-tailed test
Significance level: α = 0.05
- Willing to accept 5% chance of false rejection
Test type: One-sample t-test (testing a mean)
Data collection plan: Randomly survey students and ask them to track their study hours for a week
What this test tells you: Whether the average study time is consistent with 15 hours or not
What this test does NOT tell you: Why students study that amount, what causes variation in study time, whether study time relates to grades

Example 2: Testing a Proportion

Scenario: A politician claims that 65% of city residents support a new public transportation proposal. You want to verify this claim.

Step-by-step:

Research question: Do 65% of city residents support the proposal?
Population: All residents of the city
Variable: Support for proposal (yes or no - categorical data)
Parameter: p (population proportion who support)
Null hypothesis: H₀: p = 0.65
- Interpretation: Exactly 65% of all city residents support the proposal
Alternative hypothesis: H₁: p ≠ 0.65
- Interpretation: The true proportion is different from 65% (could be higher or lower)
- Type: Two-tailed test
Significance level: α = 0.05
- Standard significance level for this type of study
Test type: One-sample proportion test (testing a proportion)
Data collection plan: Randomly survey city residents and ask if they support the proposal (yes/no)
What this test tells you: Whether 65% support is consistent with what you observe in your sample
What this test does NOT tell you: Why people support or don’t support it, what would change their minds, how their opinion relates to other factors

Example 3: One-tailed Test

Scenario: A battery manufacturer claims their batteries last more than 20 hours on average. You want to test if this claim is true.

Step-by-step:

Research question: Do the batteries last more than 20 hours on average?
Population: All batteries produced by this manufacturer
Variable: Battery life in hours (numerical data)
Parameter: μ (population mean battery life)
Null hypothesis: H₀: μ = 20 hours
- Interpretation: The batteries last exactly 20 hours on average
- Note: The null is still “equals” even though the claim says “more than”
Alternative hypothesis: H₁: μ > 20 hours
- Interpretation: The batteries last MORE than 20 hours on average
- Type: One-tailed test (upper tail)
- Why one-tailed: The manufacturer specifically claims “more than,” so we’re only interested in testing if the mean is greater
Significance level: α = 0.05
Test type: One-sample t-test (testing a mean)
Data collection plan: Test a random sample of batteries and record how long each lasts
What this test tells you: Whether there’s evidence that average battery life exceeds 20 hours
What this test does NOT tell you: Why batteries last as long as they do, what factors affect battery life, how battery life relates to usage patterns

Quick Checklist Before Finalizing

Before you submit your hypotheses, verify each item:

✓ Clear parameter: I’ve identified whether I’m testing μ (mean) or p (proportion)
✓ Null uses equals: My null hypothesis uses = (not ≠, <, or >)
✓ Alternative is appropriate: My alternative uses ≠ (or < or > if I have good reason for one-tailed)
✓ Single group: I’m describing ONE group, not comparing multiple groups
✓ No causation language: I’m not using words like “causes,” “affects,” or “influences”
✓ No relationship language: I’m not using words like “related to,” “associated with,” or “correlation”
✓ Obtainable data: I can realistically collect the data I need
✓ Correct test selected: I know whether to use a t-test (for means) or proportion test (for proportions)
✓ Proper notation: I’m using Greek letters (μ, p) not sample statistics (x̄, p̂)
✓ Specific values: I have exact numbers in my hypotheses, not vague terms
✓ Significance level chosen: I’ve selected α before collecting data (usually 0.05)

Practice Tips for Success

Start simple: Begin with straightforward questions about single group characteristics before tackling more complex scenarios
Write null first: Always formulate your null hypothesis before the alternative. This helps clarify your thinking.
Define your population clearly: Be specific about who or what you’re studying. “Students” is vague; “undergraduate students at State University” is specific.
Avoid comparing: If you find yourself wanting to compare two groups, you’re moving beyond basic hypothesis testing. Stick to describing one group for now.
Use two-tailed tests: When in doubt, use ≠ in your alternative hypothesis. It’s the safest choice for beginners.
Think about units: Always include units (hours, inches, dollars, etc.) when discussing your parameters
Question your wording: If your hypothesis sounds like it’s explaining why something happens, rewrite it to just describe what happens.
Remember the limitation: Keep reminding yourself that hypothesis testing describes characteristics but doesn’t explain causes or relationships. This is the most common mistake beginners make.

Final Thoughts

Learning to formulate proper hypotheses takes practice. The most important things to remember are:

Hypothesis testing is descriptive, not explanatory
Focus on one parameter of one group at a time
Use proper notation and be specific
Choose your significance level before collecting data
Avoid implying causation or relationships

With these principles in mind and regular practice, you’ll become confident in formulating clear, testable, and appropriate hypotheses. Remember: good hypotheses are clear, specific, testable, and appropriately limited in scope!