Back to Statistics and Probability

Higher Applications of Mathematics

Confidence intervals

Estimating population values from samples.

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 estimating a population value from sample data and interpreting uncertainty. 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

  • A confidence interval gives a plausible range for a population value.
  • A wider interval usually means more uncertainty.
  • Interpret the interval in context, not as a guarantee for one person.

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

Confidence interval output

A sample is used to estimate the mean waiting time at a clinic.

Simulated output

> t.test(waiting$minutes)
95 percent confidence interval:
  21.4  28.6
sample mean
  25.0
202530

95% CI: 21.4 to 28.6 minutes

Sample mean

25.0 min

The mean waiting time in the sample.

Interval

21.4 to 28.6 min

A plausible range for the population mean, using this method.

Confidence level

95%

Use the confidence level in the interpretation, not as a guarantee for one person.

What it means

The interval gives a range of plausible values for the population mean waiting time. It is not a range for every individual waiting time.

What to write

Using this sample, a 95% confidence interval for the mean waiting time is 21.4 to 28.6 minutes. This suggests the true mean waiting time is likely to be in that range.

Weak answer: 95% of people waited between 21.4 and 28.6 minutes.

Watch out

Remember that a confidence interval for a mean is not the spread of individual data values.

What does the interval describe?

Choose an option, then check the feedback.

Worked examples

Worked example 1

Choose the method

A sample of households estimates average weekly spending on public transport.

  1. Identify the sample statistic and margin of error.
  2. Subtract and add the margin of error.
  3. Write the lower and upper limits with units.

The interval communicates both estimate and uncertainty.

Worked example 2

Carry out and interpret

A sample of households estimates average weekly spending on public transport.

  1. Interpret the interval as a plausible range for the population mean or proportion.
  2. Mention the confidence level if given.
  3. Avoid saying every individual value lies in the interval.

Correct interpretation is a key Higher Applications skill.

Worked example 3

Check the conclusion

A sample of households estimates average weekly spending on public transport.

  1. Compare two intervals by looking at overlap and context.
  2. Use cautious language.
  3. State whether the evidence suggests a difference.

Confidence intervals support decisions but do not remove uncertainty.

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: Confidence intervals 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