4.6.2 Statistical hypothesis testing (A-level only) - Revision Guide

What you'll learn

How to write null and alternative hypotheses for an unknown proportion.
How to choose between a lower-tailed, upper-tailed and two-tailed test.
How to calculate exact probabilities using a binomial distribution.
How to make and interpret a conclusion in the context of the question.

The setting: testing a proportion

A population proportion, denoted by $p$ , is the unknown probability that a randomly selected member of a population has a particular property.

For example, $p$ might represent:

the proportion of customers who prefer a new product;
the probability that a manufactured component is defective;
the proportion of voters supporting a proposal.

You collect a random sample of size $n$ and count the number of successes. This count is represented by the random variable $X$ .

If the trials are independent and each has the same probability of success, then:

X∼B⁡(n,p)X\sim\operatorname{B}(n,p)

where $B⁡(n,p)\operatorname{B}(n,p)$ is the binomial distribution.

Definition

Test statistic

The test statistic is the numerical result used to carry out the test. For a binomial proportion test, it is usually $X$ , the number of successes in the sample.

Statistical hypotheses

A statistical hypothesis is a statement about an unknown population parameter. Here, the parameter is the population proportion $p$ .

Every test begins with two competing hypotheses.

Definition

Null and alternative hypotheses

The null hypothesis, written $H_0$ , is the claim being tested. It normally states that the population proportion has a specified value: $H_0:p=p_0$ .
The alternative hypothesis, written $H_1$ , describes the change or difference for which you are looking.

The value $p_0$ is the claimed proportion under the null hypothesis. It gives the binomial distribution used to calculate probabilities:

X∼B⁡(n,p0)X\sim\operatorname{B}(n,p_0)

Example

Writing hypotheses

A company claims that 30% of its customers choose an annual subscription. A researcher believes that the true proportion is greater than 30%.

Let $p$ be the population proportion of customers who choose an annual subscription.
The company's claim provides the null hypothesis:
$H_0:p=0.30$
The researcher is looking specifically for an increase, so the alternative hypothesis is:
$H_1:p>0.30$

Common Mistake

Hypotheses about the sample

Write hypotheses about the population proportion $p$ , not about the observed sample proportion. The sample result is evidence used to test the hypotheses.

Choosing the correct tail

The alternative hypothesis determines which results count as evidence against $H_0$ .

Upper-tailed test

Use an upper-tailed test when you are investigating whether the proportion has increased:

H_1:p>p_0

Unusually large values of $X$ support this alternative.

Lower-tailed test

Use a lower-tailed test when you are investigating whether the proportion has decreased:

H_1:p<p_0

Unusually small values of $X$ support this alternative.

Two-tailed test

Use a two-tailed test when you are investigating whether the proportion has changed in either direction:

H1:p≠p0H_1:p\neq p_0

Both unusually small and unusually large values of $X$ provide evidence against $H_0$ .

Key Idea

The alternative chooses the tail

The wording of the investigation determines $H_1$ , and $H_1$ determines which tail probability you calculate. Do not choose the tail after seeing the sample result.

Example

Choosing the direction of a test

A machine is expected to produce 4% defective components.

To investigate whether the defect rate has fallen, use $H_1:p<0.04$ , so the test is lower-tailed.
To investigate whether the defect rate has risen, use $H_1:p>0.04$ , so the test is upper-tailed.
To investigate whether the defect rate has changed, use $H1:p≠0.04H_1:p\neq0.04$ , so the test is two-tailed.

Significance level and critical regions

The significance level, denoted by $α\alpha$ , is the maximum probability you are willing to allow for rejecting $H_0$ when it is actually true.

A test at the 5% significance level uses $α=0.05\alpha=0.05$ .

Definition

Critical region

The critical region is the set of test-statistic values that cause you to reject $H_0$ . Its probability under $H_0$ must be no greater than the significance level.

For an upper-tailed test, the critical region usually has the form $X≥cX\ge c$ . Choose the smallest integer $c$ for which:

P(X≥c)≤αP(X\ge c)\le\alpha

For a lower-tailed test, it usually has the form $X≤cX\le c$ . Choose the largest integer $c$ for which:

P(X≤c)≤αP(X\le c)\le\alpha

Because the binomial distribution is discrete, the actual probability of the critical region is often less than $α\alpha$ .

Example

Finding an upper critical region

Suppose $X∼B⁡(20,0.2)X\sim\operatorname{B}(20,0.2)$ under $H_0$ . Find the upper critical region for a 5% test.

Search for the smallest boundary $c$ such that $P(X≥c)≤0.05P(X\ge c)\le0.05$ .
A calculator gives:
$P(X≥7)≈0.0867P(X\ge7)\approx0.0867$
This is greater than 0.05, so $X≥7X\ge7$ would reject $H_0$ too frequently.
The next possible boundary gives:
$P(X≥8)≈0.0321P(X\ge8)\approx0.0321$
This is no greater than 0.05.
Therefore, the critical region is:
$X≥8X\ge8$
Its actual significance level is approximately 3.21%.

For a two-tailed test at significance level $α\alpha$ , the significance is usually shared between the two tails. Each tail is chosen to have probability no greater than $α2\frac{\alpha}{2}$ .

Example

Finding a two-tailed critical region

Let $X∼B⁡(20,0.5)X\sim\operatorname{B}(20,0.5)$ under $H_0$ . Find a two-tailed critical region at the 5% significance level.

Allocate at most 2.5% to each tail:
$α2=0.025\frac{\alpha}{2}=0.025$
In the lower tail:
$P(X≤5)≈0.0207,P(X≤6)≈0.0577P(X\le5)\approx0.0207,\qquad P(X\le6)\approx0.0577$
Therefore the lower critical region is $X≤5X\le5$ .
By symmetry:
$P(X≥15)≈0.0207P(X\ge15)\approx0.0207$
Therefore the upper critical region is $X≥15X\ge15$ .
The complete critical region is:
$X≤5orX≥15X\le5\quad\text{or}\quad X\ge15$
Its total probability is approximately $0.0207 + 0.0207 = 0.0414$ , which is below 0.05.

The probability-value method

A probability value, often called a p-value, measures how likely it would be to obtain a result at least as extreme as the observed one, assuming $H_0$ is true.

For an observed value $x$ :

upper-tailed test: calculate $P(X≥x)P(X\ge x)$ ;
lower-tailed test: calculate $P(X≤x)P(X\le x)$ ;
two-tailed test: account for extreme results in both tails.

The decision rule is:

H0.\begin{aligned} \text{p-value}\le\alpha &\Rightarrow \text{reject }H_0,\\ \text{p-value}>\alpha &\Rightarrow \text{do not reject }H_0. \end{aligned}

Tip

Include the observed result

For an upper-tail calculation use $P(X≥x)P(X\ge x)$ , not $P (X > x)$ . For a lower-tail calculation use $P(X≤x)P(X\le x)$ , not $P (X < x)$ .

Conducting a complete test

A fair coin should land heads with probability 0.5. The coin is tossed 20 times and lands heads 16 times. Test at the 5% significance level whether the coin is biased towards heads.

Example

Testing whether a coin favours heads

Let $p$ be the probability that the coin lands heads. The hypotheses are:
$H0:p=0.5,H1:p>0.5H_0:p=0.5,\qquad H_1:p>0.5$
Under $H_0$ , the number of heads has distribution:
$X∼B⁡(20,0.5)X\sim\operatorname{B}(20,0.5)$
Since $H_1:p>0.5$ , calculate the upper-tail probability:
$P(X≥16)=∑r=1620(20r)(0.5)r(0.5)20−r≈0.00591P(X\ge16)=\sum_{r=16}^{20}\binom{20}{r}(0.5)^r(0.5)^{20-r} \approx0.00591$
Compare this with the significance level:
$0.00591 < 0.05$
Therefore, reject $H_0$ .
There is sufficient evidence at the 5% significance level to suggest that the coin is biased towards heads.

Common Mistake

Accepting the null hypothesis

If the result is not significant, say there is insufficient evidence to reject $H_0$ . Do not say that $H_0$ has been proved or accepted: the sample may simply be too small to detect a real difference.

Interpreting the conclusion

Your conclusion must contain three parts:

the decision about $H_0$ ;
the strength of evidence at the stated significance level;
the conclusion in the original context.

A statistically significant result does not automatically mean that the difference is large or practically important. It means that the observed result would be sufficiently unlikely under $H_0$ .

Key Idea

Evidence, not proof

A hypothesis test measures the strength of sample evidence. Rejecting $H_0$ does not prove $H_1$ , and failing to reject $H_0$ does not prove that $H_0$ is true.

Exam technique

In the exam

Define $p$ in context and write both hypotheses using $p$ , including equality in $H_0$ .
Use $H_1$ to select the correct tail before calculating any probability.
State the distribution under $H_0$ , including both parameters.
Include the observed value in the tail probability and compare the result explicitly with $α\alpha$ .
Finish with “sufficient evidence” or “insufficient evidence” and refer directly to the claim in the question.

Self review

Check yourself

When would you use $H1:p≠p0H_1:p\neq p_0$ rather than a one-tailed alternative?
If $P(X≥x)=0.073P(X\ge x)=0.073$ in a 5% upper-tailed test, what decision should you make?
Why can the actual significance level of an exact binomial test be less than the stated significance level?

What you'll learn

How to write null and alternative hypotheses for an unknown proportion.
How to choose between a lower-tailed, upper-tailed and two-tailed test.
How to calculate exact probabilities using a binomial distribution.
How to make and interpret a conclusion in the context of the question.

The setting: testing a proportion

A population proportion, denoted by $p$ , is the unknown probability that a randomly selected member of a population has a particular property.

For example, $p$ might represent:

the proportion of customers who prefer a new product;
the probability that a manufactured component is defective;
the proportion of voters supporting a proposal.

You collect a random sample of size $n$ and count the number of successes. This count is represented by the random variable $X$ .

If the trials are independent and each has the same probability of success, then:

X∼B⁡(n,p)X\sim\operatorname{B}(n,p)

where $B⁡(n,p)\operatorname{B}(n,p)$ is the binomial distribution.

Definition

Test statistic

The test statistic is the numerical result used to carry out the test. For a binomial proportion test, it is usually $X$ , the number of successes in the sample.

Statistical hypotheses

A statistical hypothesis is a statement about an unknown population parameter. Here, the parameter is the population proportion $p$ .

Every test begins with two competing hypotheses.

Definition

Null and alternative hypotheses

The null hypothesis, written $H_0$ , is the claim being tested. It normally states that the population proportion has a specified value: $H_0:p=p_0$ .
The alternative hypothesis, written $H_1$ , describes the change or difference for which you are looking.

The value $p_0$ is the claimed proportion under the null hypothesis. It gives the binomial distribution used to calculate probabilities:

X∼B⁡(n,p0)X\sim\operatorname{B}(n,p_0)

Example

Writing hypotheses

A company claims that 30% of its customers choose an annual subscription. A researcher believes that the true proportion is greater than 30%.

Let $p$ be the population proportion of customers who choose an annual subscription.
The company's claim provides the null hypothesis:
$H_0:p=0.30$
The researcher is looking specifically for an increase, so the alternative hypothesis is:
$H_1:p>0.30$

Common Mistake

Hypotheses about the sample

Write hypotheses about the population proportion $p$ , not about the observed sample proportion. The sample result is evidence used to test the hypotheses.

Choosing the correct tail

The alternative hypothesis determines which results count as evidence against $H_0$ .

Upper-tailed test

Use an upper-tailed test when you are investigating whether the proportion has increased:

H_1:p>p_0

Unusually large values of $X$ support this alternative.

Lower-tailed test

Use a lower-tailed test when you are investigating whether the proportion has decreased:

H_1:p<p_0

Unusually small values of $X$ support this alternative.

Two-tailed test

Use a two-tailed test when you are investigating whether the proportion has changed in either direction:

H1:p≠p0H_1:p\neq p_0

Both unusually small and unusually large values of $X$ provide evidence against $H_0$ .

Key Idea

The alternative chooses the tail

The wording of the investigation determines $H_1$ , and $H_1$ determines which tail probability you calculate. Do not choose the tail after seeing the sample result.

Example

Choosing the direction of a test

A machine is expected to produce 4% defective components.

To investigate whether the defect rate has fallen, use $H_1:p<0.04$ , so the test is lower-tailed.
To investigate whether the defect rate has risen, use $H_1:p>0.04$ , so the test is upper-tailed.
To investigate whether the defect rate has changed, use $H1:p≠0.04H_1:p\neq0.04$ , so the test is two-tailed.

Significance level and critical regions

The significance level, denoted by $α\alpha$ , is the maximum probability you are willing to allow for rejecting $H_0$ when it is actually true.

A test at the 5% significance level uses $α=0.05\alpha=0.05$ .

Definition

Critical region

The critical region is the set of test-statistic values that cause you to reject $H_0$ . Its probability under $H_0$ must be no greater than the significance level.

For an upper-tailed test, the critical region usually has the form $X≥cX\ge c$ . Choose the smallest integer $c$ for which:

P(X≥c)≤αP(X\ge c)\le\alpha

For a lower-tailed test, it usually has the form $X≤cX\le c$ . Choose the largest integer $c$ for which:

P(X≤c)≤αP(X\le c)\le\alpha

Because the binomial distribution is discrete, the actual probability of the critical region is often less than $α\alpha$ .

Example

Finding an upper critical region

Suppose $X∼B⁡(20,0.2)X\sim\operatorname{B}(20,0.2)$ under $H_0$ . Find the upper critical region for a 5% test.

Search for the smallest boundary $c$ such that $P(X≥c)≤0.05P(X\ge c)\le0.05$ .
A calculator gives:
$P(X≥7)≈0.0867P(X\ge7)\approx0.0867$
This is greater than 0.05, so $X≥7X\ge7$ would reject $H_0$ too frequently.
The next possible boundary gives:
$P(X≥8)≈0.0321P(X\ge8)\approx0.0321$
This is no greater than 0.05.
Therefore, the critical region is:
$X≥8X\ge8$
Its actual significance level is approximately 3.21%.

For a two-tailed test at significance level $α\alpha$ , the significance is usually shared between the two tails. Each tail is chosen to have probability no greater than $α2\frac{\alpha}{2}$ .

Example

Finding a two-tailed critical region

Let $X∼B⁡(20,0.5)X\sim\operatorname{B}(20,0.5)$ under $H_0$ . Find a two-tailed critical region at the 5% significance level.

Allocate at most 2.5% to each tail:
$α2=0.025\frac{\alpha}{2}=0.025$
In the lower tail:
$P(X≤5)≈0.0207,P(X≤6)≈0.0577P(X\le5)\approx0.0207,\qquad P(X\le6)\approx0.0577$
Therefore the lower critical region is $X≤5X\le5$ .
By symmetry:
$P(X≥15)≈0.0207P(X\ge15)\approx0.0207$
Therefore the upper critical region is $X≥15X\ge15$ .
The complete critical region is:
$X≤5orX≥15X\le5\quad\text{or}\quad X\ge15$
Its total probability is approximately $0.0207 + 0.0207 = 0.0414$ , which is below 0.05.

The probability-value method

A probability value, often called a p-value, measures how likely it would be to obtain a result at least as extreme as the observed one, assuming $H_0$ is true.

For an observed value $x$ :

upper-tailed test: calculate $P(X≥x)P(X\ge x)$ ;
lower-tailed test: calculate $P(X≤x)P(X\le x)$ ;
two-tailed test: account for extreme results in both tails.

The decision rule is:

H0.\begin{aligned} \text{p-value}\le\alpha &\Rightarrow \text{reject }H_0,\\ \text{p-value}>\alpha &\Rightarrow \text{do not reject }H_0. \end{aligned}

Tip

Include the observed result

For an upper-tail calculation use $P(X≥x)P(X\ge x)$ , not $P (X > x)$ . For a lower-tail calculation use $P(X≤x)P(X\le x)$ , not $P (X < x)$ .

Conducting a complete test

A fair coin should land heads with probability 0.5. The coin is tossed 20 times and lands heads 16 times. Test at the 5% significance level whether the coin is biased towards heads.

Example

Testing whether a coin favours heads

Let $p$ be the probability that the coin lands heads. The hypotheses are:
$H0:p=0.5,H1:p>0.5H_0:p=0.5,\qquad H_1:p>0.5$
Under $H_0$ , the number of heads has distribution:
$X∼B⁡(20,0.5)X\sim\operatorname{B}(20,0.5)$
Since $H_1:p>0.5$ , calculate the upper-tail probability:
$P(X≥16)=∑r=1620(20r)(0.5)r(0.5)20−r≈0.00591P(X\ge16)=\sum_{r=16}^{20}\binom{20}{r}(0.5)^r(0.5)^{20-r} \approx0.00591$
Compare this with the significance level:
$0.00591 < 0.05$
Therefore, reject $H_0$ .
There is sufficient evidence at the 5% significance level to suggest that the coin is biased towards heads.

Common Mistake

Accepting the null hypothesis

Interpreting the conclusion

Your conclusion must contain three parts:

the decision about $H_0$ ;
the strength of evidence at the stated significance level;
the conclusion in the original context.

A statistically significant result does not automatically mean that the difference is large or practically important. It means that the observed result would be sufficiently unlikely under $H_0$ .

Key Idea

Evidence, not proof

A hypothesis test measures the strength of sample evidence. Rejecting $H_0$ does not prove $H_1$ , and failing to reject $H_0$ does not prove that $H_0$ is true.

Exam technique

In the exam

Define $p$ in context and write both hypotheses using $p$ , including equality in $H_0$ .
Use $H_1$ to select the correct tail before calculating any probability.
State the distribution under $H_0$ , including both parameters.
Include the observed value in the tail probability and compare the result explicitly with $α\alpha$ .
Finish with “sufficient evidence” or “insufficient evidence” and refer directly to the claim in the question.

Self review

Check yourself

When would you use $H1:p≠p0H_1:p\neq p_0$ rather than a one-tailed alternative?
If $P(X≥x)=0.073P(X\ge x)=0.073$ in a 5% upper-tailed test, what decision should you make?
Why can the actual significance level of an exact binomial test be less than the stated significance level?

2.5.2 Statistical hypothesis testing

What you'll learn

The setting: testing a proportion

Statistical hypotheses

Choosing the correct tail

Upper-tailed test

Lower-tailed test

Two-tailed test

Significance level and critical regions

The probability-value method

Conducting a complete test

Interpreting the conclusion

Review 4.6.2 Statistical hypothesis testing (A-level only) by teaching Genie

2.5.2 Statistical hypothesis testing

What you'll learn

The setting: testing a proportion

Statistical hypotheses

Choosing the correct tail

Upper-tailed test

Lower-tailed test

Two-tailed test

Significance level and critical regions

The probability-value method

Conducting a complete test

Interpreting the conclusion

Review 4.6.2 Statistical hypothesis testing (A-level only) by teaching Genie

2.5.2 Statistical hypothesis testing

What you'll learn

The setting: testing a proportion

Statistical hypotheses

Choosing the correct tail

Upper-tailed test

Lower-tailed test

Two-tailed test

Significance level and critical regions

The probability-value method

Conducting a complete test

Interpreting the conclusion

Review 4.6.2 Statistical hypothesis testing (A-level only) by teaching Genie

4.6.2 Statistical hypothesis testing (A-level only) Revision Guide

2.5.2 Statistical hypothesis testing

What you'll learn

The setting: testing a proportion

Statistical hypotheses

Choosing the correct tail

Upper-tailed test

Lower-tailed test

Two-tailed test

Significance level and critical regions

The probability-value method

Conducting a complete test

Interpreting the conclusion

Review 4.6.2 Statistical hypothesis testing (A-level only) by teaching Genie

4.6.2 Statistical hypothesis testing (A-level only) Revision Guide