Confidence Interval for the Difference Between Means Calculator for Unknown Population Variances


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

Sample Mean (\(\bar X_1\)):
Sample Mean (\(\bar X_2\)):
Sample St. Dev. \(s_1\):
Sample St. Dev. \(s_2\):
Sample Size (\(n_1\)):
Sample Size (\(n_2\)):
Confidence Level (Ex: 0.95, 95, 99, 99%) =
Assume equal variances
Assume unequal variances
Test for equality of variances



Confidence Interval for the Difference Between Means Calculator

The use of Confidence intervals extends beyond estimating specific parameters, as it can also be used for operations between parameters. In this specific case, the objective is to construct a confidence interval (CI) for the difference between two population means (\(\mu_1 - \mu_2\)), in the case that the population standard deviation are not known, in which case the expression for the confidence interval is:

\[ CI = \left(\bar X_1 - \bar X_2 - t_c \times \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, \bar X_1 - \bar X_2 + t_c \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right) \]

when the population variances are assumed to be unequal, and

\[ CI = \left(\bar X_1 - \bar X_2 - t_c \times s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, \bar X_1 - \bar X_2 + t_c \times s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right) \]

when the population variances are assumed to be equal. The critical t-value correspond to critical values associated to the t-distribution, and the number of degrees of freedom depend on whether the population variances are equal or unequal. The number of degrees of freedom for equal population variances is \(df = n_1 + n_2 - 2\), and the number of degrees of freedom

Assumptions that need to be met

In this case, as with majority of parametric procedures, we need to have that the samples come from normally distributed populations. In this case we don't have to assume that the population standard deviations are known (which is a more realistic assumption than the case where it is assumed that they are known).

More Confidence Interval Calculators

Observe that if you do know both population standard deviations, you will want to use the calculator for the confidence interval of the difference between means for known population variances. For one mean only use this calculator..




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