# Hypothesis Tests (S2)

> Edexcel International A-Level Mathematics · IAL S2 2018
> Source: https://www.owlsprep.com/study/edexcel-ial-math-s2-hypothesis-tests/

This guide covers all Edexcel IAL S2 hypothesis test content, including core sampling concepts, hypothesis formulation, critical regions, one/two-tailed tests, and exams for binomial $p$ and Poisson mean $λ$, with normal approximation for binomial tests.

**Prerequisites:** [Understanding of binomial and Poisson discrete probability distributions](https://www.owlsprep.com/study/edexcel-ial-math-s2-discrete-probability-distributions/); [Ability to use Edexcel cumulative statistical tables](https://www.owlsprep.com/study/edexcel-ial-math-s2-statistical-tables-use/)

## Learning objectives

- Distinguish between population, census, sample, sampling unit and sampling frame, and compare advantages/disadvantages of census vs sample surveys
- Define a statistic and explain the concept of its sampling distribution
- Formulate null and alternative hypotheses for binomial $p$ and Poisson mean $λ$ tests
- Identify critical regions and calculate test statistics for one-tailed and two-tailed tests
- Apply normal approximation with continuity correction for binomial $p$ hypothesis tests where appropriate
- Write exam-standard contextual conclusions distinguishing between rejecting $H_0$ and accepting $H_1$

## Core Sampling and Statistic Fundamentals

Before conducting hypothesis tests, you must first distinguish between key sampling terms defined in the Edexcel S2 specification. A **population** is the entire set of items of interest, a **census** collects data from every member of the population, while a **sample** collects data from a subset of the population. A **sampling unit** is an individual member of the population, and a **sampling frame** is the list of all sampling units available for selection.

**Statistic** — A function of known observations from a sample that contains no unknown population parameters. Its sampling distribution is the probability distribution of the statistic across all possible samples of the same size from the population.

- Census: Advantage = 100% population coverage, no sampling error; Disadvantage = time-consuming, expensive, impractical for large or destructively tested populations
- Sample survey: Advantage = fast, low-cost, suitable for destructive testing; Disadvantage = sampling error present, results are estimates not exact values

**Worked example:** A factory produces 10,000 lightbulbs per day. The quality control team wants to test what proportion of bulbs last longer than 1000 hours. State one advantage and one disadvantage of using a sample survey instead of a census for this test.

1. Identify context: Testing bulb lifespan is destructive, so testing every bulb would make them unsellable.
2. Advantage of sample: Avoids destroying all produced bulbs, reducing significant financial loss for the factory.
3. Disadvantage of sample: The estimated proportion of long-life bulbs will have sampling error, so it may not perfectly match the true population value.

> **Exam tip:** Examiners regularly ask 1-2 mark comparison questions on census vs sample, so tailor your answers to the context given in the question, not just generic statements.

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## Hypothesis Test Fundamentals & Hypothesis Formulation

A hypothesis test is a statistical procedure to test a claim about a population parameter using sample data. You will always state two competing hypotheses: the null hypothesis ($H_0$) and alternative hypothesis ($H_1$).

**Hypothesis Test** — A structured test to assess whether there is sufficient evidence from sample data to reject a stated assumption (null hypothesis) about a population parameter.

- $H_0$ is the default assumption, stated strictly as an equality: $H_0: p = p_0$ or $H_0: λ = λ_0$
- $H_1$ is the claim you are testing for evidence to support, stated as an inequality matching the test direction
- Significance level $α$ is the threshold probability of incorrectly rejecting $H_0$ when it is true, usually 1%, 5% or 10% in S2 exams

**Worked example:** A teacher claims that 70% of her students pass their mock exam. A new tutor believes the pass rate is lower. Formulate suitable null and alternative hypotheses for a test of the tutor's claim.

1. Define the parameter explicitly: Let $p$ = probability a randomly selected student passes the mock exam.
2. Null hypothesis is the original default claim, stated as an equality:
3. $$H_0: p = 0.7$$
4. Alternative hypothesis matches the tutor's claim of a lower pass rate, so it is a lower tail inequality:
5. $$H_1: p < 0.7$$

> **Exam tip:** Always define your parameter explicitly when formulating hypotheses, as this is a mandatory mark point in all S2 hypothesis test questions.

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## Critical Regions, One-Tailed and Two-Tailed Tests

The test statistic is the sample statistic you calculate to assess the evidence against $H_0$. The **critical region** is the set of values of the test statistic for which you reject $H_0$. The type of test (one-tailed or two-tailed) is determined by the $H_1$ inequality.

**Critical Region** — The range of test statistic values where the probability of observing the result under $H_0$ is less than or equal to the significance level $α$, leading to rejection of $H_0$.

- One-tailed test: $H_1$ has a single direction inequality (< or >), so all of the $α$ probability is allocated to one tail of the distribution
- Two-tailed test: $H_1$ has a non-directional inequality (≠), so the $α$ probability is split equally between the upper and lower tails of the distribution, e.g. 2.5% in each tail for a 5% significance level

**Worked example:** A shop claims that the average number of customer complaints per week is 3. A manager believes this value has changed, with no specified direction of change. State the type of test required, and the critical region allocation for a 5% significance level.

1. Define the parameter: Let $λ$ = mean number of complaints per week, following a Poisson distribution.
2. Hypotheses are $H_0: λ = 3$, $H_1: λ \neq 3$, so this is a two-tailed test.
3. For a 5% significance level, allocate 2.5% probability to the lower tail and 2.5% to the upper tail of the Poisson distribution under $H_0$ to find the critical region boundaries.

> **Exam tip:** For two-tailed tests, you must explicitly state that you split the significance level equally between both tails, as this is a common mark loss point.

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## Hypothesis Tests for Binomial Parameter $p$

The most common S2 hypothesis test is for the parameter $p$ of a binomial distribution $X \sim B(n,p)$. You can use cumulative binomial tables to find critical regions or p-values, or use a normal approximation when $np$ and $n(1-p)$ are both ≥5, with mandatory continuity correction.

**Worked example:** A coin is tossed 20 times, and lands on heads 6 times. Test at the 5% significance level whether the coin is biased towards tails.

1. Define parameter: Let $p$ = probability the coin lands on heads, $X$ = number of heads in 20 tosses, so $X \sim B(20, p)$ under $H_0$.
2. Formulate hypotheses: Biased towards tails means $p < 0.5$, so $H_0: p = 0.5$, $H_1: p < 0.5$, one-tailed test, $α = 0.05$.
3. Test statistic is $x=6$. Use cumulative binomial tables for $n=20, p=0.5$ to find $P(X \leq 6)$:
4. $$P(X \leq 6) = 0.0577$$
5. Compare to significance level: 0.0577 > 0.05, so the test statistic is not in the critical region.
6. Conclusion: There is insufficient evidence at the 5% significance level to reject $H_0$, so we conclude the coin is not biased towards tails.

**Worked example:** A company claims 40% of customers buy a supplementary product. In a sample of 100 customers, 30 buy the product. Test at the 5% significance level whether the true proportion is lower than 40%, using normal approximation.

1. Define $X \sim B(100, 0.4)$ under $H_0$. Check normal approximation conditions: $np = 40$, $n(1-p) = 60$, both ≥5, so approximation is suitable.
2. Approximate distribution: $X \approx N(40, 24)$ (variance = $np(1-p) = 100 \times 0.4 \times 0.6 = 24$).
3. Apply continuity correction for $X \leq 30$: use $x = 30.5$.
4. Calculate z-score:
5. $$z = \frac{30.5 - 40}{\sqrt{24}} = -1.936$$
6. Critical value for 5% one-tailed lower test is -1.6449. $-1.936 < -1.6449$, so test statistic is in the critical region.
7. Conclusion: Reject $H_0$ at 5% significance level, there is sufficient evidence the proportion of customers buying the supplementary product is lower than 40%.

> **Exam tip:** Always apply continuity correction when using normal approximation for binomial hypothesis tests, as missing this will cost you 2+ marks per question.

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## Hypothesis Tests for Poisson Mean $λ$

You will also test hypotheses for the mean $λ$ of a Poisson distribution, used for count data (e.g. number of accidents per month, number of defects per batch). Use cumulative Poisson tables from the provided formula booklet to find p-values or critical regions.

**Worked example:** A café records an average of 4 customers arriving per 10 minutes. After a new advertising campaign, 9 customers arrive in a 10 minute window. Test at the 5% significance level whether the arrival rate has increased.

1. Define $X$ = number of customers arriving per 10 minutes, under $H_0$ $X \sim Po(4)$. Hypotheses: $H_0: λ = 4$, $H_1: λ > 4$, one-tailed test, $α = 0.05$.
2. Test statistic is $x=9$. Find $P(X \geq 9) = 1 - P(X \leq 8)$ using Poisson cumulative tables for $λ=4$:
3. $$P(X \leq 8) = 0.9786 \implies P(X \geq 9) = 1 - 0.9786 = 0.0214$$
4. Compare to significance level: 0.0214 < 0.05, so test statistic is in the critical region.
5. Conclusion: Reject $H_0$ at 5% significance level, there is sufficient evidence the customer arrival rate has increased after the advertising campaign.

> **Exam tip:** For Poisson upper tail tests, remember to subtract the cumulative lower tail probability from 1 to get the p-value, as tables only provide cumulative lower tail values.

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## Common pitfalls

- **Wrong:** Stating $H_1$ as an equality instead of an inequality
  - Why it fails: The null hypothesis is always the default equality assumption; $H_1$ is the competing claim which must be an inequality.
  - Correct: Write $H_0: p = 0.6$ and $H_1: p < 0.6$ for a lower tail test, never $H_1: p = 0.4$ or similar.
- **Wrong:** Forgetting continuity correction when using normal approximation for binomial tests
  - Why it fails: Binomial is a discrete distribution and normal is continuous, so continuity correction is required to align the two distributions.
  - Correct: Adjust discrete values by 0.5, e.g. use 30.5 for $X \leq 30$, 29.5 for $X \geq 30$.
- **Wrong:** Splitting the significance level equally for a one-tailed test
  - Why it fails: One-tailed tests allocate all significance level probability to a single tail, so no split is needed.
  - Correct: Use the full $α$ value for one tail, and split $α/2$ per tail only for two-tailed tests.
- **Wrong:** Writing conclusions that state "accept $H_0$" instead of "insufficient evidence to reject $H_0$"
  - Why it fails: You never prove the null hypothesis is true, only that there is not enough evidence to refute it.
  - Correct: Explicitly state "There is insufficient evidence at the X% significance level to reject $H_0$" when the test statistic is not in the critical region.
- **Wrong:** Failing to define the population parameter when formulating hypotheses
  - Why it fails: Edexcel examiners require explicit parameter definition as a mandatory mark point for all hypothesis test questions.
  - Correct: Always start hypothesis questions with "Let $p$ = probability of success on a single trial" or equivalent for Poisson $λ$.

## Cheatsheet

| Concept | Key Rule | Exam Check |
| --- | --- | --- |
| Hypothesis Formulation | $H_0$ = equality, $H_1$ = inequality matching claim | Did you define your parameter explicitly? |
| One-tailed Test | All $α$ allocated to one tail, $H_1$ has < or > | Is the claim directional? Use one-tailed only if yes |
| Two-tailed Test | $α$ split equally between two tails, $H_1$ has ≠ | Did you split $α$ if test is non-directional? |
| Binomial $p$ Test | Use cumulative binomial tables; normal approx if $np,n(1-p)≥5$ | Did you apply continuity correction for normal approx? |
| Poisson $λ$ Test | Use cumulative Poisson tables; upper tail = 1 - $P(X≤x)$ | Did you subtract from 1 for upper tail probabilities? |
| Conclusion | Reference significance level, contextualize result | Did you distinguish 'reject $H_0$' vs 'insufficient evidence to reject $H_0$'? |

## What's next

Now that you have mastered S2 hypothesis tests, you can apply these skills to past paper questions to build exam speed and accuracy. Hypothesis testing is a core statistical skill that you will build on if you study S3, where you will cover z-tests for normal means, chi-squared goodness of fit tests, and correlation hypothesis tests. Make sure you practice using the official Edexcel statistical tables to find critical values and p-values quickly, as this will save you valuable time in the exam. Remember to always structure your answers clearly with parameter definition, hypotheses, test statistic calculation, comparison to critical value or significance level, and contextual conclusion to maximize your mark gain.

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