Back to Statistics and Probability

Higher Applications of Mathematics

Hypothesis testing

Testing claims with evidence from data.

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Statistics reference

Check probability, diagrams, averages, regression, intervals and tests.

Mixed revision quiz

Review Statistics and Probability with adaptive feedback.

Before you start

  • Be confident reading values from tables and graphs.
  • Check units, sample size and what each variable represents.
  • Use context in written answers, especially when interpreting results.

Method chooser

Which statistics method do I use?

Are there stages or repeated chance events?

Use a tree diagram.

Do categories overlap?

Use a Venn diagram.

Do you need a typical value?

Calculate an average.

Do you need to compare consistency?

Compare spread using standard deviation or IQR.

Are you comparing distributions?

Use box plots or histograms.

Are you checking a relationship?

Use scatter graph and correlation.

Do you need a prediction line?

Use linear regression.

Are you estimating a population value?

Use a confidence interval.

Are you judging evidence against a claim?

Use a hypothesis test.

Are the same people measured twice?

Use a paired t-test.

Are you comparing two proportions?

Use a z-test for two proportions.

Statistics lesson

Key idea

  • This topic focuses on using sample evidence, p-values and significance levels to test claims. In Higher Applications, the aim is to use statistical methods to make careful decisions from real data.
  • Good statistical work has three parts: choose a suitable method, carry it out accurately, then explain what the result means in the situation.
  • When writing conclusions, use cautious language such as 'this suggests' or 'there is evidence to suggest'. Data can support a conclusion, but it rarely proves it completely.

Key formulae, definitions and methods

  • The null hypothesis is the default claim being tested.
  • The alternative hypothesis is the claim supported if evidence is strong enough.
  • If p-value <= significance level, reject the null hypothesis.

Technology output practice

Interpreting statistical output

Read the simulated output, pick out the key value, then turn it into a written conclusion. This is a learning preview, not a real RStudio environment.

Context

Hypothesis test output

A paired test compares typing speeds before and after a short training session.

Simulated output

> t.test(after, before, paired = TRUE)
t = 2.41, df = 17, p-value = 0.028
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
  1.2  8.7

p-value = 0.028

0.028 is less than 0.05

Statistically significant evidence at the 5% level

p-value

0.028

Less than 0.05, so there is statistically significant evidence at the 5% level.

Test type

Paired

The same people were measured before and after.

Conclusion

Evidence of change

Use evidence language rather than proof language.

What it means

Because p = 0.028 is below 0.05, the output supports a statistically significant change in mean typing speed at the 5% level.

What to write

At the 5% significance level, there is statistically significant evidence that the training changed mean typing speed. The paired design is appropriate because the same people were measured before and after.

Weak answer: The training is proven to work for everyone.

Watch out

Do not use proof language. A small p-value supports evidence for a difference; it does not prove the claim for every person.

At the 5% level, what should you decide from p = 0.028?

Choose an option, then check the feedback.

Worked examples

Worked example 1

Choose the method

A local authority tests whether a new form has reduced average processing time.

  1. State the null and alternative hypotheses in context.
  2. Identify the significance level, such as 5%.
  3. Use the p-value from technology or a supplied output.

The method turns sample evidence into a structured decision.

Worked example 2

Carry out and interpret

A local authority tests whether a new form has reduced average processing time.

  1. Compare the p-value with the significance level.
  2. Decide whether to reject or fail to reject the null hypothesis.
  3. Write the conclusion in plain English.

The p-value comparison controls the conclusion.

Worked example 3

Check the conclusion

A local authority tests whether a new form has reduced average processing time.

  1. Avoid saying the result proves the claim.
  2. Mention evidence and uncertainty.
  3. Connect the decision back to the real situation.

A good conclusion uses evidence language rather than proof language.

Watch out

  • Choosing a method because it is familiar rather than because it matches the data.
  • Giving a numerical answer without explaining what it means in context.
  • Mixing up sample evidence with certainty about the whole population.
  • Ignoring outliers, skewness, units or the scale on a graph.
  • Using causal language when the data only shows association.

Technology connection

Related RStudio and Spreadsheet topics

RStudio: Hypothesis testing in RStudio

Next step

Move into practice

Use the learning notes to choose suitable summaries and conclusions, then try varied data sets, tables, p-values and interpretation prompts.

Statistics mixed quiz