Introduction
In this article we will discuss step by step procedure to construct a confidence interval for difference between two population means when the population standard deviations are known.
CI for difference between two population means (variances are known)
Let $X_1, X_2, \cdots, X_{n_1}$ be a random sample of size $n_1$ from a population with mean $\mu_1$ and standard deviation $\sigma_1$.
Let $Y_1, Y_2, \cdots, Y_{n_2}$ be a random sample of size $n_2$ from a population with mean $\mu_2$ and standard deviation $\sigma_2$. And the two sample are independent.
Let $\overline{X} = \frac{1}{n_1}\sum X_i$ and $\overline{Y} =\frac{1}{n_2}\sum Y_i$ be the sample means of first and second sample respectively.
Let $C=1-\alpha$ be the confidence coefficient. Our objective is to construct a $100(1-\alpha)$% confidence interval estimate for the difference $(\mu_1-\mu_2)$.
The margin of error for the difference of means is
$$ \begin{aligned} E &= Z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} \end{aligned} $$
where $Z_{\alpha/2}$ is the value from normal statistical table for desired confidence coefficient.
Thus $100(1-\alpha)$% confidence interval estimate for the difference $(\mu_1-\mu_2)$ is $$ \begin{aligned} (\overline{X} -\overline{Y})- E \leq (\mu_1-\mu_2) \leq (\overline{X} -\overline{Y}) + E. \end{aligned} $$
Assumptions
a. The two samples are independent.
b. Both the samples are simple random sample.
c. Both the samples comes from population having normal distribution.
d. The two population variances $\sigma^2_1$ and $\sigma^2_2$ are known.
Step by Step Procedure
Step by step procedure to estimate the confidence interval for difference between two population means is as follows:
Step 1 Specify the confidence level $(1-\alpha)$
Step 2 Specify the given information
Given that sample sizes $n_1, n_2$, samples means $\overline{X},\overline{Y}$, standard deviations $\sigma_1, \sigma_2$.
Step 3 Specify the formula
$100(1-\alpha)$% confidence interval estimate for the difference $(\mu_1-\mu_2)$ is
$$ \begin{aligned} (\overline{X} -\overline{Y})- E \leq (\mu_1-\mu_2) \leq (\overline{X} -\overline{Y}) + E. \end{aligned} $$
where $E = Z_{\alpha/2}\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$.
Step 4 Determine the critical value
Find the critical value $Z_{\alpha/2}$ from the normal statistical table for desired confidence level.
Step 5 Compute the margin of error
The margin of error for the difference of means is
$$ \begin{aligned} E = Z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} \end{aligned} $$
Step 6 Determine the confidence interval
$100(1-\alpha)$% confidence interval estimate for the difference $(\mu_1-\mu_2)$ is
$$ \begin{aligned} (\overline{X} -\overline{Y})- E \leq (\mu_1-\mu_2) \leq (\overline{X} -\overline{Y}) + E. \end{aligned} $$
That is $100(1-\alpha)$% confidence interval estimate for the difference $(\mu_1-\mu_2)$ is $(\overline{X} -\overline{Y})\pm E$ or $\big((\overline{X} -\overline{Y})- E, (\overline{X} -\overline{Y})+E\big)$.
