Skip to content

setting up the hypothesis

At the heart of every hypothesis test are two competing statements about a population:

  • The null hypothesis H_0
  • The alternative hypothesis H_a

They must be:
Mutually exclusive (can’t both be true)
Exhaustive (cover all possibilities)
About population parameters, not sample statistics

Let’s break down how to set them up correctly.

🎯 1. Start with the research question

Ask yourself: What am I trying to show, detect, or test?

  • If you want to show a difference → H_a contains the difference
  • If you want to show an increase → H_a contains “>”
  • If you want to show a decrease → H_a contains “<”

The alternative hypothesis always reflects the researcher’s suspicion or claim.

🎯 2. The null hypothesis is the “no effect” or “no difference” statement

It usually states:

  • equality
  • no change
  • no relationship
  • no improvement

Examples:

  • H_0: \mu = 50
  • H_0: p = 0.40
  • H_0: \mu_1 = \mu_2

The null is the default assumption — the thing you try to find evidence against.

🎯 3. Choose the correct form of the alternative hypothesis

There are three types:

A. Two‑tailed test (difference in either direction)

Use when you care about any difference.

H_a: \mu \neq \mu_0

Example:
“Is the average score different from 70?”

B. Right‑tailed test (greater than)

Use when you want to show an increase.

H_a: \mu > \mu_0

Example:
“Does the new fertilizer increase plant height?”

C. Left‑tailed test (less than)

Use when you want to show a decrease.

H_a: \mu < \mu_0

Example:
“Is the average waiting time shorter than 10 minutes?”

🎯 4. Make sure hypotheses are about population parameters

Correct:

  • H_0: \mu = 100
  • H_a: p > 0.30

Incorrect:

  • H_0: \bar{x} = 100
  • H_a: \hat{p} > 0.30

Sample statistics belong in the test statistic, not the hypotheses.

🎯 5. Keep the null hypothesis with equality

The null always includes the equality sign:

  • =
  • \le
  • \ge
See also  Bonferroni correction - What It Is and Why It Matters

The alternative never includes equality.

⭐ Fresh, Original Examples

Example 1: Testing a Mean

Claim: The average battery life is 10 hours.

H_0: \mu = 10 \\ H_a: \mu \neq 10

Example 2: Testing for an Increase

A teacher believes a new method improves test scores.

H_0: \mu_{\text{new}} \le \mu_{\text{old}} \\ H_a: \mu_{\text{new}} > \mu_{\text{old}}

Example 3: Testing a Proportion

A company claims 40% of customers choose the premium plan.

H_0: p = 0.40 \\ H_a: p \neq 0.40

Example 4: Comparing Two Groups

Do men and women have different average daily step counts?

H_0: \mu_{\text{men}} = \mu_{\text{women}} \\ H_a: \mu_{\text{men}} \neq \mu_{\text{women}}

Example 5: Paired Test (Before/After)

Does a training program reduce reaction time?

H_0: \mu_d \ge 0 \\H_a: \mu_d < 0

(where \mu_d is the mean of the differences)

🎨 The Big Idea

Setting up hypotheses is about:

  • translating a research question into math
  • deciding what “no effect” looks like
  • deciding what kind of effect you’re testing for

Once the hypotheses are set, the rest of the test flows naturally.

Leave a Reply

error: Content is protected !!