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.
WHAT YOU PROBABLY THINK
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. Since the samples are independent, the difference of the two means is normal as well, and the variances add:
Under the null hypothesis of equal means the second bracket is zero, leaving a plain z score. The standard errors are combined by adding their squares, never by subtracting: the opening claim would make the spread of a difference smaller than the spread of one 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. By the Central Limit Theorem each sample mean is approximately normal whatever the population shape, and the sample variances s² are close enough to the population ones to be used in their place. The statistic is the same with s² replacing σ², and the conclusion is approximate rather than exact.
A confidence interval for the difference follows from the same standard error, and it answers the same question as the test: if the interval excludes zero, a two-tailed test at the matching level rejects equality. Quoting the interval as well as the test result is worth doing: it says how large the difference might be, where the test only says that one exists.
YOUR TURN
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, so the size of the difference remains poorly pinned down even though its existence is established.
THE EXAM BIT
- 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 rather than exact.
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
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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.
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.
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