## 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 $\big((\overline{X} -\overline{Y})- E, (\overline{X} -\overline{Y})+E\big)$.