Inferential Statistics • Lesson 12
Null & Alternative Hypotheses
Learn how statistical hypotheses turn a real-world claim into a testable statement about a population.
What You Will Learn
Understand the purpose of statistical hypotheses.
Distinguish between the null and alternative hypotheses.
Write hypotheses using population parameters.
Identify two-sided, left-tailed, and right-tailed alternatives.
Translate business questions into statistical hypotheses.
Avoid common mistakes when writing H₀ and Hₐ.
Why Do We Need Hypotheses?
In real-world analytics, we often want to test a claim. For example, a company may claim that its average delivery time is 30 minutes.
We usually cannot measure every delivery in the population. Instead, we collect a sample and use statistical inference to evaluate whether the sample provides enough evidence against the claim.
Population statement
Collect evidence
Evaluate the claim
STEP 1
Start With a Population Claim
Suppose a food-delivery company claims:
Company claim
μ = 30 minutes
Here, μ represents the true population mean delivery time.
We can use hypotheses to formally represent this claim and the possibility that the claim is not correct.
STEP 2
The Two Hypotheses
H₀
μ = 30
The null hypothesis represents the reference claim or baseline assumption that we test against.
Hₐ
μ ≠ 30
The alternative hypothesis represents the pattern we want to find evidence for instead of the null hypothesis.
STEP 3
Why Is Equality Usually in H₀?
In the standard hypothesis-testing setup, the null hypothesis contains the equality because it provides a specific reference value for calculating how unusual the sample result would be.
H₀
μ = 30
Hₐ
μ ≠ 30
Evidence
Sample data
STEP 4
Three Common Forms of Hₐ
| Test Type | Alternative | Meaning |
|---|---|---|
| Two-sided | μ ≠ μ₀ | The population value may be higher or lower. |
| Right-tailed | μ > μ₀ | The population value is greater. |
| Left-tailed | μ < μ₀ | The population value is smaller. |
Two-Sided Test
Use a two-sided alternative when the question is whether the population parameter is different from the reference value in either direction.
H₀: μ = 50
Hₐ: μ ≠ 50
Example question: “Has the average delivery time changed from 50 minutes?”
Right-Tailed Test
Use a right-tailed alternative when the research question is specifically asking whether the population value is greater.
H₀: μ = 50
Hₐ: μ > 50
Example question: “Is the average delivery time greater than 50 minutes?”
Left-Tailed Test
Use a left-tailed alternative when the research question is specifically asking whether the population value is smaller.
H₀: μ = 50
Hₐ: μ < 50
Example question: “Has the average delivery time fallen below 50 minutes?”
STEP 5
Translate Business Questions Into Hypotheses
Question 1: Is the average customer spending different from ₹2,000?
H₀: μ = ₹2,000
Hₐ: μ ≠ ₹2,000
Question 2: Is the average delivery time greater than 30 minutes?
H₀: μ = 30
Hₐ: μ > 30
Question 3: Is the defect rate below 5%?
H₀: p = 0.05
Hₐ: p < 0.05
Population Parameter vs Sample Statistic
Hypotheses describe the population parameter. The sample statistic is then used as evidence when testing the hypothesis.
| Concept | Example |
|---|---|
| Population mean | μ |
| Sample mean | x̄ |
| Population proportion | p |
| Sample proportion | p̂ |
Key Idea
Think of H₀ as the Reference Model
The hypothesis test asks whether the observed sample evidence would be sufficiently unusual if the null hypothesis were the reference assumption.
H₀
Reference assumption
Sample
Observed evidence
Test
Quantifies how unusual the evidence is
Common Mistakes
❌ Putting the equality only in Hₐ
In the standard setup, equality is placed in H₀.
❌ Using the sample statistic as the hypothesis
Hypotheses are statements about population parameters, not simply observed sample values.
❌ Choosing the tail after seeing the data
The direction of the alternative should come from the research question and be specified before analyzing the result.
❌ Saying “fail to reject” means H₀ is proven
Failing to reject H₀ means the evidence was not sufficient to reject it under the chosen testing procedure.
PRACTICE
Test Your Understanding
CHECK YOUR UNDERSTANDING
Which hypothesis represents the reference claim being tested?
CHECK YOUR UNDERSTANDING
If the research question asks whether μ is different from 100, what should Hₐ be?
CHECK YOUR UNDERSTANDING
Which alternative represents a right-tailed test?
CHECK YOUR UNDERSTANDING
A company wants to know whether its average delivery time is below 30 minutes. Which Hₐ is appropriate?
CHECK YOUR UNDERSTANDING
Which symbol normally represents the population proportion?
CHECK YOUR UNDERSTANDING
If H₀: μ = 500 and Hₐ: μ ≠ 500, what type of test is this?
Fill in the Blank
The null hypothesis is commonly written as H__.
The alternative hypothesis is commonly written as H__.
If Hₐ: μ > 40, the test is ______-tailed.
If Hₐ: μ < 40, the test is ______-tailed.
Analytics Challenge
Turn a Business Question Into Hypotheses
An online store claims that its average order value is ₹2,500. An analyst wants to determine whether the true average order value is different from ₹2,500.
Your task:
Write H₀ and Hₐ.
H₀: μ = ₹2,500
Hₐ: μ ≠ ₹2,500
Because the question asks whether the value is “different,” the alternative must allow both higher and lower values.
Final Challenge
Choose the Correct Alternative
A website team wants to know whether its average page-load time has increased above the target of 2.5 seconds.
What should the alternative hypothesis be?
Hₐ: μ > 2.5 seconds
The word “increased” tells us that the research question is directional, so this is a right-tailed alternative.
Lesson 12 Summary
H₀ is the null hypothesis and acts as the reference assumption.
Hₐ represents the alternative claim we want evidence for.
Hypotheses describe population parameters such as μ or p.
Hₐ: μ ≠ μ₀ creates a two-sided test.
Hₐ: μ > μ₀ creates a right-tailed test.
Hₐ: μ < μ₀ creates a left-tailed test.
The direction of the test should come from the research question, not from the observed result.
Failing to reject H₀ does not mean H₀ has been proven true.