Comparing two groups and making the call
The difference confidence interval
- Compare two groups by bootstrapping the difference in medians, e.g. (A − B).
- This gives a 95% confidence interval for the difference between the two population medians — a plausible range for that difference.
Making the formal call
- Look at whether the interval contains 0:
- Entirely above 0 → be confident that in the population, A's median is greater than B's.
- Entirely below 0 → be confident that B's median is greater.
- Contains 0 → you cannot make a call; the data are consistent with no difference.
- State the call in context, naming the populations and the variable — not just "A is bigger".
Justifying and limiting the call
- Justify with the interval: "the whole interval is above 0, so I am confident the difference is positive."
- Scope: the call generalises to the populations only if the samples were random samples from them.
- Note limitations: small samples give wide intervals, and bias in how the data were collected can undermine the conclusion.
Comparing weekly screen time (hours) of Year 13 boys and girls, a 95% bootstrap confidence interval for (boys − girls) is to hours. Make and justify a formal inference.
Step 1 — Check the interval against 0
The interval to is entirely above 0.
Step 2 — Make the call, in context
I am confident that back in the population, the median weekly screen time of Year 13 boys is greater than that of Year 13 girls.
Step 3 — Justify and scope
Justify: the whole confidence interval for (boys − girls) is positive, so the difference is confidently greater than 0. Scope: this holds for the populations these samples represent, provided the students were randomly sampled; otherwise it applies only to the groups sampled.
A 95% bootstrap CI for the difference in median test score (Class A − Class B) is to marks. State the conclusion.
Step 1 — Check against 0
The interval runs from to , so it contains 0.
Step 2 — Conclusion
I cannot make a call about which class has the greater population median test score — a difference of 0 is plausible, so the data do not show one class scoring higher. A larger sample would give a narrower interval that might settle the question.
Test yourself
Practice by grade
One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.
A 95% bootstrap CI for the difference in median mass (Group A − Group B) is to kg. Can a call be made, and if so, which group has the greater median?
A 95% bootstrap CI for the difference in median rainfall (Region P − Region Q, mm) is to mm. Make and justify a formal inference in context.
A researcher compares median incomes of two suburbs from a convenience sample of people at a shopping mall. The 95% bootstrap CI for the difference is to dollars. Write a full conclusion, addressing whether a call can be made and any limitations.