Difference between two means (known variances)

The distribution

For independent random samples of sizes nx and ny from N(μx, σx²) and N(μy, σy²):

X̄ - Ȳ ~ N(μx - μy, σx²/nx + σy²/ny).

The variances add (it is a difference of independent variables), each divided by its own sample size.

The test

H0: μx = μy (or μx - μy = d), H1: μx > μy, μx < μy or μx ≠ μy. Define the subscripts.

Test statistic: z = ((x̄ - ȳ) - (μx - μy)) / sqrt(σx²/nx + σy²/ny), where μx - μy is the value in H0 (often 0).

Compare with the percentage points (4 d.p.) and conclude in context.

Confidence interval for the difference

(x̄ - ȳ) ± z sqrt(σx²/nx + σy²/ny).

If the interval contains 0, there is no evidence (at that level, two-tailed) of a difference between the means.

Assumptions: the two samples are independent random samples and the populations are normal (or the samples are large).

Worked example

Plot yields with fertiliser A are N(μA, 2.4²) and with B are N(μB, 2.1²). Samples of 30 and 25 plots have means 21.6 and 20.4. Test at 5% whether A gives a higher mean yield.

1. H0: μA = μB, H1: μA > μB.
2. Standard error = sqrt(2.4²/30 + 2.1²/25) = 0.6070
3. z = 1.2/0.6070 = 1.9771
4. 1.977 > 1.6449: reject H0.

Answer: There is evidence that fertiliser A gives a higher mean yield than B.

Check yourself

1. Samples of 30 and 25 are taken from N(μA, 2.4²) and N(μB, 2.1²). Find the standard error of X̄A - X̄B. Give your answer to 3 significant figures.
   Show answer

   0.607. sqrt(5.76/30 + 4.41/25) = 0.6070.

2. Samples of 30 and 25 from N(μA, 2.4²) and N(μB, 2.1²) have means 21.6 and 20.4. Calculate the test statistic for H0: μA = μB. Give your answer to 3 significant figures.
   Show answer

   1.98. 1.2/0.6070 = 1.9771.

3. Samples of 30 and 25 from N(μA, 2.4²) and N(μB, 2.1²) have means 21.6 and 20.4. Find the lower limit of a 95% confidence interval for μA - μB. Give your answer to 2 decimal places.
   Show answer

   0.01. 1.2 - 1.96 × 0.6070 = 0.0104.

4. With the same data, test H0: μA - μB = 1 against H1: μA - μB > 1. Calculate the test statistic. Give your answer to 3 significant figures.
   Show answer

   0.330. (1.2 - 1)/0.6070 = 0.3295.

5. H0: μA = μB, H1: μA > μB at the 5% level gives z = 1.977. Which conclusion is correct?
   • Reject H0: there is evidence that the mean yield with A is greater than with B
   • Do not reject H0: the yields are the same
   • Reject H0: the yields are the same
   • Do not reject H0: A gives a greater yield
   Show answer

   Reject H0: there is evidence that the mean yield with A is greater than with B. 1.977 > 1.6449, so significant; conclude in context with the direction.

6. A 95% confidence interval for μx - μy is (-0.4, 2.1). What does this suggest?
   • μx is definitely greater than μy
   • μx - μy = 0.85 exactly
   • There is no evidence at the 5% level of a difference between the means
   • The samples were not independent
   Show answer

   There is no evidence at the 5% level of a difference between the means. 0 is inside the interval, so a difference of 0 is plausible.

7. Which is a required assumption for this test with small samples?
   • Each population is normally distributed
   • The two samples have the same size
   • The sample means are equal
   • The two populations have the same mean
   Show answer

   Each population is normally distributed. With small samples, normal populations are needed for X̄ - Ȳ to be normal; the samples must also be independent.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3
• Pearson Edexcel IAL Statistics S3 (WST03) mark scheme, June 2024
• Pearson Edexcel Mathematical Formulae and Statistical Tables

Last updated 8 November 2026. Spotted a mistake? Tell us.