Inference with normal and t-distributions
Choose and carry out one- and two-sample inference for means, including pooled variance and confidence intervals.
Exam scope and assessment notes
Mapped to Cambridge International AS & A Level Further Mathematics 9231. These original explanations and self-checks are independent study support, not Cambridge questions, marks or grades. The syllabus assumes prior knowledge of the relevant 9709 Mathematics content; check each component's prerequisites with your teacher. Full-paper assessment and human subject-specialist review remain outstanding.
Before you begin
- Cambridge expects learners to know the whole Cambridge International A Level Mathematics (9709) content. Paper 1 further pure mathematics draws on 9709 Papers 1 and 3. Confirm your exam year and component route with your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What makes a Further Mathematics solution convincing?
What you will learn
- Formulate hypotheses and use a one-sample t-test
- Select between two-sample, paired t-tests and normal tests
- Estimate pooled variance and construct confidence intervals for means or differences
Choose the test from the sampling design
For a small sample from a normal population with unknown variance, use a t statistic. A one-sample test compares a sample mean with a hypothesised population mean; the degrees of freedom are n−1.
For two independent samples, a pooled t-test assumes a common population variance. For paired data, first calculate within-pair differences and test their mean. A normal test is appropriate when the variance is known or the syllabus conditions support it.
Hypotheses and intervals must match the question
State H₀ and H₁ before calculating. The alternative can be one-sided or two-sided, and determines the tail area. Compare the calculated statistic with a critical value or use the p-value at the stated significance level.
A confidence interval is estimate ± critical value × standard error. For a difference of means, choose the t or normal distribution as appropriate; do not interpret failure to reject as proof that the means are equal.
Pause and explain
Which habit best checks your work on inference with normal and t-distributions?
Worked example
A normal population has unknown variance. A sample of 10 has mean 18 and standard deviation 3. Test H₀: μ=16 against H₁: μ>16 and calculate t.
Show the worked solution
- Use t=(x̄−μ₀)/(s/√n).
- t=(18−16)/(3/√10)=2√10/3≈2.11.
- There are 9 degrees of freedom; compare with the 5% one-sided critical value if that level is specified.
Answer t≈2.11 with 9 degrees of freedom; a decision requires the chosen significance level and critical value.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A sample of 16 values from a normal population has mean 24 and standard deviation 4. Calculate the one-sample t statistic for H₀: μ=22.
Give me a hint
Use the sample standard error s/√n.
Compare my reasoning
- The standard error is 4/√16=1.
- t=(24−22)/1.
- The degrees of freedom are 15.
t=2 with 15 degrees of freedom.
Look for these in your work
- Standard error correct
- Statistic calculated
- Degrees of freedom stated
Two independent normal samples have n₁=8, s₁²=9 and n₂=10, s₂²=16. Find the pooled estimate of the common variance.
Give me a hint
Weight each sample variance by its degrees of freedom.
Compare my reasoning
- sₚ²=[(n₁−1)s₁²+(n₂−1)s₂²]/(n₁+n₂−2).
- Substitute: [7(9)+9(16)]/16.
- The numerator is 63+144=207.
sₚ²=207/16=12.9375.
Look for these in your work
- Degrees-of-freedom weights correct
- Denominator correct
- Value calculated
A researcher measures the same 12 learners before and after tutoring. Which t-test design is appropriate and what values should be tested?
Give me a hint
Measurements from the same learners are paired, not independent.
Compare my reasoning
- Calculate one difference for each learner, using a consistent before/after order.
- Test the mean of those 12 differences against zero or the specified change.
- Use a paired t-test with 11 degrees of freedom if its normal-difference assumptions are appropriate.
Use a paired t-test on the within-learner differences, with 11 degrees of freedom.
Look for these in your work
- Pairing recognised
- Differences defined
- Degrees of freedom and assumption noted
Common mistakes
- Using a two-sided critical value for a one-sided alternative
- Using independent-sample methods on paired observations
- Treating failure to reject H₀ as proof the null is true
What can you explain now?
What degrees of freedom are used for a one-sample t-test with n observations?
Compare with the explanation
n−1
One sample mean has been estimated from the data.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
How do sample design, variance information and the alternative hypothesis determine which inference procedure to use?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.