Maths › Further Statistics 2 › Comparing two normal means
Comparing two normal means
Two samples, two means, and a question about whether the populations behind them differ. The difference of the sample means is itself normal, and everything follows from its standard error.
Builds on Combinations of normal random variables and Estimators, standard error and confidence intervals.
IN THIS TOPIC
- Find the standard error of a difference of sample means.
- Carry out a two-sample z test and state the conclusion in context.
- Build a confidence interval for the difference and link it to the test.
COMMON MISCONCEPTION
To compare two sample means, subtract their standard errors to get the standard error of the difference.
The difference has its own distribution
Each sample mean is normal about its population mean, with variance σ²/n. The two samples are independent, so the difference of the means is normal as well, and the variances add:
It is printed in the booklet under Sampling distributions, among the tests for the mean when σ is known, so read it off and substitute. Hypotheses are about the population means, so write H₀: μx = μy. Under that null the second bracket in the numerator vanishes and you are left with a plain z score. Standard errors combine by adding their squares, never by subtracting. Subtraction would make the spread of a difference smaller than the spread of a single mean, which is the wrong way round.
WORKED EXAMPLE
A two-sample test
Sample 1: n = 40, mean 52.3, from a population with variance 25. Sample 2: n = 50, mean 49.8, variance 36. Test at 5% whether the population means differ.
H0: μx = μy; H1: μx ≠ μy. Two-tailed at 5%, so the critical values are ±1.96.
Standard error = √(25/40 + 36/50) = √1.345 = 1.160.
z = 2.5/1.160 = 2.16. Since 2.16 > 1.96, reject H0. There is evidence at the 5% level of a difference between the population means.
When the variances are unknown
Large samples rescue the method twice over. The Central Limit Theorem makes each sample mean approximately normal whatever the population shape, and the sample variances s² are close enough to the population values to stand in for them. The statistic is unchanged with s² replacing σ², and the conclusion becomes approximate. Say so, because a mark usually hangs on it.
A confidence interval for the difference follows from the same standard error, and it answers the same question as the test. An interval excluding zero corresponds exactly to a two-tailed test at the matching level rejecting equality. Quoting the interval as well as the verdict is worth doing, since it says how large the difference might be where the test says only that one exists.
GUIDED PRACTICE
An interval for the difference
For the samples above, find a 95% confidence interval for the difference in population means, and comment.
Show the working
The point estimate is 52.3 − 49.8 = 2.5, and the standard error is 1.160 as before.
The interval is 2.5 ± 1.96 × 1.160 = 2.5 ± 2.27, that is (0.23, 4.77).
Zero lies outside, so the two-tailed test at 5% rejects equality, agreeing with the test already carried out.
The interval is wide. The size of the difference stays poorly pinned down even though its existence is established.
ASSESSMENT FOCUS
- Set out the standard error as a separate calculation, showing both variance-over-n terms.
- Say whether the test is one-tailed or two-tailed before quoting a critical value.
- State the independence of the two samples. The variance rule depends on it.
- With unknown variances, say that large samples make the result approximate.
CHECK YOURSELF
Two independent samples of size 25 have means 80 and 74, from populations with variances 50 and 30. Find the standard error of the difference.
Show a hint
Divide each variance by its own n, add, then take the root.
Show the answer
50/25 + 30/25 = 2 + 1.2 = 3.2, so the standard error is √3.2 = 1.789 to three decimal places.
The difference of two independent sample means is normal, with the two variance-over-n terms added under the square root.
With large samples the sample variances may replace the population ones, and an interval missing zero matches a significant two-tailed test.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the comparing two normal means questions page.
CHECK YOUR PROGRESS
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- Find the standard error of a difference of sample means.
- Carry out a two-sample z test and state the conclusion in context.
- Build a confidence interval for the difference and link it to the test.
Open the full revision checklist to see every objective in the course in one place.