Tests for the difference between two Normal meansEdexcel A-Level Further Maths: Revision notes

Section 1

The distribution of the difference of two sample means

Take independent random samples $X_1,\ldots,X_{n_x}$ from $\mathrm{N}(\mu_x,\sigma_x^2)$ and $Y_1,\ldots,Y_{n_y}$ from $\mathrm{N}(\mu_y,\sigma_y^2)$ . Then $\bar X\sim\mathrm{N}\left(\mu_x,\frac{\sigma_x^2}{n_x}\right)$ and $\bar Y\sim\mathrm{N}\left(\mu_y,\frac{\sigma_y^2}{n_y}\right)$ . The difference of independent Normal variables is Normal, with the means subtracted but the variances added: $\bar X-\bar Y\sim\mathrm{N}\left(\mu_x-\mu_y,\ \frac{\sigma_x^2}{n_x}+\frac{\sigma_y^2}{n_y}\right).$

Key termsdifference of meansindependent samples

Common mistake

Subtracting the variances. Variances always add for a difference of independent variables.

Section 2

The test with known variances

Under $H_0:\mu_x=\mu_y$ , the test statistic is $Z=\frac{(\bar X-\bar Y)-(\mu_x-\mu_y)}{\sqrt{\frac{\sigma_x^2}{n_x}+\frac{\sigma_y^2}{n_y}}}\sim\mathrm{N}(0,1),$ with $\mu_x-\mu_y=0$ under $H_0$ . Method: 1. state $H_0$ and $H_1$ in terms of population means; 2. calculate $z$ ; 3. compare with the critical value, or find the $p$ -value; 4. conclude in context. Example: $n_x=20$ , $\sigma_x^2=9$ , $\bar x=52.8$ ; $n_y=30$ , $\sigma_y^2=16$ , $\bar y=51.2$ . Standard error $=\sqrt{0.45+0.5333}=0.992$ , $z=\frac{1.6}{0.992}=1.61$ .

Key termstest statisticnull hypothesis

Common mistake

Writing hypotheses with $\bar x$ and $\bar y$ . Hypotheses are always about population means $\mu_x$ and $\mu_y$ .

Section 3

One-tailed and two-tailed tests

Choose the alternative from the wording. 'Differ' or 'is different' gives two-tailed $H_1:\mu_x\ne\mu_y$ , with the significance level split between both tails (critical values $\pm1.96$ at $5\%$ ). 'Greater' or 'larger' gives one-tailed $H_1:\mu_x>\mu_y$ (critical value $1.645$ at $5\%$ , $2.326$ at $1\%$ ). For a two-tailed test double the tail probability to get the $p$ -value. Reject $H_0$ if $p<$ the significance level. Always write the conclusion in context, as evidence for or against, never as proof: 'there is insufficient evidence that ...' rather than 'the means are the same'.

Key termsone-tailedtwo-tailedp-value

Exam tip

Look for direction words in the question before choosing $H_1$ . If there is no direction, use a two-tailed test.

Section 4

Unknown variances and large samples

When the population variances are unknown, replace $\sigma_x^2$ and $\sigma_y^2$ by the sample variances $s_x^2$ and $s_y^2$ . For large samples the Central Limit Theorem says $\bar X-\bar Y$ is approximately Normal, even if the populations are not Normal, and $s^2$ is close to $\sigma^2$ , so $Z=\frac{(\bar X-\bar Y)-(\mu_x-\mu_y)}{\sqrt{\frac{S_x^2}{n_x}+\frac{S_y^2}{n_y}}}\approx\mathrm{N}(0,1).$ Example: $n_1=60$ , $s_1^2=18.4$ , $\bar x_1=24.6$ ; $n_2=75$ , $s_2^2=22.5$ , $\bar x_2=23.1$ . Standard error $=\sqrt{0.3067+0.3}=0.779$ , $z=\frac{1.5}{0.779}=1.93<1.96$ , so a two-tailed $5\%$ test does not reject $H_0$ ( $p=0.054$ ).

Key termsCentral Limit Theoremlarge sample

Exam tip

Quote the Central Limit Theorem and the large samples when you use sample variances in place of population variances.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Tests for the difference between two Normal means

Independent random samples are taken from two Normal populations. Sample $X$ : $n_x=20$ , $\bar x=52.8$ , population variance $\sigma_x^2=9$ . Sample $Y$ : $n_y=30$ , $\bar y=51.2$ , population variance $\sigma_y^2=16$ . Both variances are known.
Test, at the $5\%$ significance level, whether the mean of $X$ is greater than the mean of $Y$ . State your conclusion in context.2 marks
Reaction times, in milliseconds, are measured for two independent groups, $A$ and $B$ . Each reaction time is Normally distributed with standard deviation $40$ . Group $A$ has $36$ people with mean time $312$ . Group $B$ has $49$ people with mean time $298$ . A researcher tests whether the mean reaction times of the two populations differ, at the $5\%$ level.
State the conclusion of the test, with a reason, in context.2 marks
Two classes take the same task. The times, in minutes, are not assumed to be Normal and the population variances are unknown. Class $1$ : $n_1=60$ , $\bar x_1=24.6$ , $s_1^2=18.4$ . Class $2$ : $n_2=75$ , $\bar x_2=23.1$ , $s_2^2=22.5$ . The teacher tests whether the population mean times differ.
Calculate the value of the test statistic for $H_0:\mu_1=\mu_2$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).