# Confidence Interval for the Difference Between Means Calculator

Instructions: Use this step-by-step calculator for a confidence interval for the difference between two Means, for known population variances, by providing the sample data in the form below:

Sample mean 1 $$(\bar X_1)$$ =
Population Standard Deviation 1 $$(\sigma_1)$$
Sample Size 1 $$(N_1)$$
Sample mean 2 $$(\bar X_2)$$ =
Population Standard Deviation 2 $$(\sigma_2)$$
Sample Size 2 $$(N_2)$$
Confidence Level (Ex: 0.95, 95, 99, 99%) =

## Confidence Interval for the Difference Between Means Calculator

Confidence intervals can be used not only for a specific parameter, but also for operations between parameters. In this specific case, we are interested in constructing a confidence interval for the difference between two population means ($$\mu_1 - \mu_2$$), the following expression for the confidence interval is used:

$CI = \left(\bar X_1 - \bar X_2 - z_c \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}, \bar X_1 - \bar X_2 + z_c \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}\right)$

where in this case the critical value correspond to critical values associated to the Normal distribution. The critical values for the given $$\alpha$$ is $$z_c = z_{1 - \alpha/2}$$.

### Assumptions that need to be met

In this specific case, we need to have that the samples come from normally distributed populations, and it is assumed that the population standard deviations are known (which is a somewhat unrealistic assumption, but it is sometimes met).

### More Confidence Interval Calculators

Observe that if you do not know both population standard deviations, you will want to use the calculator for the confidence interval of the difference between means for unknown population variances. For one mean only use this calculator..

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