Hypothesis Testing Fundamentals

Master the language and concepts of statistical hypothesis testing

SPEC 5.1 • STATISTICAL HYPOTHESIS TESTING

What is Hypothesis Testing?

Definition

Hypothesis testing is a statistical method for making decisions about a population based on sample data.

We use hypothesis testing when we want to determine whether there is enough evidence to support a claim about a population parameter.

The Logic

We start by assuming something is true (our default position), then collect data to see if there's sufficient evidence to reject this assumption.

Real-World Examples

Quality control: Has a manufacturing process changed?

Medical trials: Is a new drug more effective than existing treatment?

Market research: Has customer satisfaction improved?

Null and Alternative Hypotheses

NULL HYPOTHESIS (H₀)

The default assumption

States that nothing has changed or there is no effect

Always contains "="

ALTERNATIVE HYPOTHESIS (H₁)

What we are testing for

States that something has changed or there is an effect

Contains <, >, or ≠

Key Point

We either reject H₀ (in favour of H₁) or do not reject H₀. We never "accept" H₀ or "prove" anything.

Important

The null hypothesis H₀ must always include the equality. For example: H₀: p = 0.3, not H₀: p > 0.3

Writing Hypotheses

Hypotheses are written in terms of population parameters:

ParameterSymbolExample H₀Used For

Population proportionpH₀: p = 0.5Binomial tests

Population meanμH₀: μ = 100Normal tests

Population correlationρH₀: ρ = 0Correlation tests

Exam Tip

Always define your parameter in words before stating your hypotheses. For example: "Let p be the probability that a customer prefers Brand A."

One-Tailed vs Two-Tailed Tests

One-Tailed Test

Tests for change in a specific directionH₁: p > 0.5 or H₁: p < 0.5Critical region at one end of distribution

Two-Tailed Test

Tests for change in either directionH₁: p ≠ 0.5Critical region at both ends of distribution

How to Identify from Context

Keywords suggesting ONE-TAILEDKeywords suggesting TWO-TAILED"increased", "improved", "greater than", "more than", "higher""changed", "different", "altered", "not equal to""decreased", "reduced", "less than", "lower", "fewer""whether there is a difference", "if there has been any change"

Visualising Test Types

CRITICAL REGIONS FOR DIFFERENT TEST TYPES

Two-Tailed Note

For a two-tailed test at significance level α, the critical region is split: α/2 in each tail. For example, at 5% significance, there is 2.5% in each tail.

Significance Level

Definition

The significance level (α) is the probability of rejecting H₀ when it is actually true.

Significance LevelAs PercentageInterpretation

α = 0.1010%Less strict, easier to reject H₀

α = 0.055%Most common level used

α = 0.011%More strict, need stronger evidence

Key Insight

A smaller significance level means we require stronger evidence to reject H₀. The significance level equals the size of the critical region.

Exam Tip

The question will always tell you the significance level to use. If not specified, assume 5%.

Test Statistic and Critical Values

Test Statistic

A value calculated from your sample data that is used to make a decision about H₀.

Examples: sample proportion, sample mean, number of successes

Critical Value

A boundary value from statistical tables that separates the critical region from the acceptance region.

Depends on: significance level and test type

Decision Rule

Compare the test statistic to the critical value:

If test statistic is in the critical region → Reject H₀

If test statistic is not in the critical region → Do not reject H₀

Critical Region and Acceptance Region

Critical Region

The set of all values of the test statistic that would lead us to reject H₀.

Also called the rejection region.

Acceptance Region

The set of all values of the test statistic that would lead us to not reject H₀.

Complements the critical region.

Example: One-tailed test (right) at 5% significance

Important

We say "do not reject H₀" rather than "accept H₀" because failing to reject doesn't prove H₀ is true—it just means we don't have enough evidence against it.

The p-value

Definition

The p-value is the probability of obtaining a result at least as extreme as the observed result, assuming H₀ is true.

Decision Rule Using p-value

If p-value < significance level → Reject H₀

If p-value ≥ significance level → Do not reject H₀

Interpreting p-values

p-valueEvidence against H₀p > 0.10Little or no evidence0.05 < p ≤ 0.10Weak evidence0.01 < p ≤ 0.05Moderate evidencep ≤ 0.01Strong evidence

Common Misconception

The p-value is NOT the probability that H₀ is true. It's the probability of getting our observed result (or more extreme) if H₀ were true.

Evaluating a Hypothesis Test

There are two equivalent methods to decide whether to reject H₀:

Method 1: Critical Value Approach

Compare the test statistic to the critical value

Find the critical value(s) that define the rejection region

If test statistic falls in the critical region → Reject H₀

If test statistic falls in the acceptance region → Do not reject H₀

Method 2: P-Value Approach

Compare the p-value to the significance level (α)

Calculate P(getting result this extreme or more extreme | H₀ true)

If p-value < α → Reject H₀

If p-value ≥ α → Do not reject H₀

Key Point

Both methods will always give the same conclusion. Use whichever the question requires or whichever you find easier. The p-value method is often preferred as it shows exactly how significant the result is.

Conducting a Hypothesis Test

Here is the complete structure for answering a hypothesis test question:

Example Problem

A coin is claimed to be fair. To test this, the coin is tossed 20 times and 15 heads are observed. Test at the 5% significance level whether the coin is biased towards heads.

Step-by-Step Solution

Step 1: Define the parameter and state hypotheses

Let p = probability of obtaining a head

H₀: p = 0.5 (coin is fair)

H₁: p > 0.5 (coin is biased towards heads)

Step 2: State the distribution under H₀

Under H₀, X ~ B(20, 0.5) where X = number of heads

Step 3: Calculate the p-value (or find critical region)

P(X ≥ 15) = 1 − P(X ≤ 14) = 1 − 0.9793 = 0.0207

Step 4: Compare and conclude

0.0207 < 0.05, so reject H₀

Step 5: Write conclusion in context

There is sufficient evidence at the 5% significance level to conclude that the coin is biased towards heads.

Worked Example01

A coin is suspected of being biased towards heads. Let p be the probability of obtaining a head when the coin is tossed. Write down suitable null and alternative hypotheses to test this claim.

1

Identify what we're testing: The claim is that the coin is biased towards heads, meaning p is greater than for a fair coin.

2

For a fair coin, p = 0.5. This is our default assumption (null hypothesis).

3

Since we're testing if the coin is biased towards heads specifically, this is a one-tailed test (p > 0.5).

AnswerH₀: p = 0.5 (the coin is fair) H₁: p > 0.5 (the coin is biased towards heads)

Worked Example02

A factory claims that the mean weight of their chocolate bars is 50g. A quality controller suspects the bars are underweight. Let μ be the population mean weight. Write down suitable hypotheses.

1

The claim is that the mean weight equals 50g. This is our null hypothesis.

2

The quality controller suspects the bars are underweight, meaning μ < 50.

3

Since we're testing for a specific direction (underweight), this is a one-tailed test.

AnswerH₀: μ = 50 (mean weight is 50g as claimed) H₁: μ < 50 (mean weight is less than 50g)

Worked Example03

A company believes that 30% of its customers prefer online shopping. After a marketing campaign, they want to test whether this proportion has changed. State suitable hypotheses.

1

Let p = the proportion of customers who prefer online shopping.

2

The original claim is that p = 0.30. This is our null hypothesis.

3

The word "changed" suggests testing for any difference—increase OR decrease. This is a two-tailed test.

AnswerH₀: p = 0.30 (proportion is unchanged at 30%) H₁: p ≠ 0.30 (proportion has changed)

Worked Example04

In a hypothesis test, the p-value is calculated to be 0.032. State whether you would reject H₀ at: (a) the 1% significance level, (b) the 5% significance level.

1

Recall: Reject H₀ if p-value < significance level.

2

(a) At 1% level (α = 0.01):

p-value = 0.032 > 0.01

Since 0.032 > 0.01, do NOT reject H₀.

3

(b) At 5% level (α = 0.05):

p-value = 0.032 < 0.05

Since 0.032 < 0.05, reject H₀.

Answer(a) Do not reject H₀ at 1% level (b) Reject H₀ at 5% level

Worked Example05

A manufacturer claims 20% of products are defective. A sample of 50 products contains 15 defective items. The p-value for testing H₀: p = 0.2 against H₁: p > 0.2 is 0.048. Interpret this result at the 5% level.

1

Compare p-value to significance level: 0.048 < 0.05

2

Since p-value < α, we reject H₀.

3

Write conclusion in context:

Note: Always state the conclusion in context of the original problem!

AnswerThere is sufficient evidence at the 5% significance level to reject the manufacturer's claim. The data suggests that the proportion of defective products is greater than 20%.

Practice

Question 1: A teacher believes that more than 60% of students prefer morning classes. Let p be the proportion who prefer morning classes. State suitable null and alternative hypotheses, and identify the type of test.

Solution:

The teacher believes the proportion is more than 60%, so this is testing for an increase (one direction).

H₀: p = 0.60 (proportion is 60%)

H₁: p > 0.60 (proportion is greater than 60%)

Type: One-tailed test (upper tail)

Practice

Question 2: For a one-tailed test at the 5% level with H₁: p < 0.3, the test statistic follows B(25, 0.3) under H₀. The observed value is X = 4. Given that P(X ≤ 3) = 0.0332 and P(X ≤ 4) = 0.0905, determine whether to reject H₀.

Solution:

For H₁: p < 0.3, we need the critical region in the lower tail.

Critical region: values where P(X ≤ c) ≤ 0.05

P(X ≤ 3) = 0.0332 < 0.05 → 3 is in critical region

P(X ≤ 4) = 0.0905 > 0.05 → 4 is NOT in critical region

Critical region: X ≤ 3

Observed value X = 4 is NOT in the critical region.

Conclusion: Do not reject H₀

Practice

Question 3: In a hypothesis test of H₀: μ = 50 against H₁: μ ≠ 50, the calculated p-value is 0.042. State the conclusion at the 5% level and at the 1% level, writing each conclusion in context.

Solution:

At 5% level: p-value = 0.042 < 0.05

Reject H₀. There is sufficient evidence at the 5% significance level to conclude that the population mean is not equal to 50.

At 1% level: p-value = 0.042 > 0.01

Do not reject H₀. There is insufficient evidence at the 1% significance level to conclude that the population mean is different from 50.

Summary

Key Terminology

TermDefinition

H₀ (Null hypothesis)Default assumption; contains equality

H₁ (Alternative hypothesis)What we test for; contains <, >, or ≠

Significance level (α)P(reject H₀ | H₀ true); typically 1%, 5%, 10%

Critical regionValues leading to rejection of H₀

p-valueP(result as extreme as observed | H₀ true)

Key Exam Skills

Always define your parameter before stating hypotheses

Use context clues to determine one-tailed vs two-tailed

Compare p-value to significance level (not to 0.5!)

Write conclusions in context of the original problem