Back to Statistics and Probability

Higher Applications of Mathematics

t-tests, paired t-tests and z-tests

Selecting and interpreting common statistical tests.

Statistics referenceStatistics mixed quiz

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 choosing and interpreting common tests for means, paired data and two proportions. 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

  • Use a t-test for a mean when the population standard deviation is not known.
  • Use a paired t-test when the same people or matched items are measured twice.
  • Use a z-test for two proportions when comparing percentages from two groups.

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 health project compares activity levels before and after a school walking challenge.

  1. Decide whether the data is one mean, paired measurements or two proportions.
  2. Choose the matching test.
  3. Use the output p-value to make the decision.

Choosing the test is as important as carrying it out.

Worked example 2

Carry out and interpret

A health project compares activity levels before and after a school walking challenge.

  1. For paired data, calculate or analyse the differences for each person.
  2. Do not treat before and after values as unrelated groups.
  3. Interpret the result as a change within the same group.

Paired tests are used when values are naturally linked.

Worked example 3

Check the conclusion

A health project compares activity levels before and after a school walking challenge.

  1. For two proportions, compare the sample percentages.
  2. Use the z-test output to judge significance.
  3. Write the conclusion in terms of the two populations.

The final sentence should identify the groups and the claim being tested.

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: t-tests and paired t-tests in RStudioRStudio: z-tests for two proportions

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