T-test for two Means – Unknown Population Standard Deviations

Instructions: This calculator conducts a t-test for two population means (\(\mu_1\) and \(\mu_2\)), with unknown population standard deviations. This test apply when you have two-independent samples, and the population standard deviations \(\sigma_1\) and \(\sigma_2\) and not known. Please select the null and alternative hypotheses, type the significance level, the sample means, the sample standard deviations, the sample sizes, and the results of the t-test for two independent samples will be displayed for you:

Ho: \(\mu_1\) \(\mu_2\)
Ha: \(\mu_1\) \(\mu_2\)
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 (\(n_2\)):
Significance Level (\(\alpha\)) =
Assume equal variances
Assume unequal variances
Test for equality of variances

The T-test for Two Independent Samples

More about the t-test for two means so you can better interpret the output presented above: A t-test for two means with unknown population variances and two independent samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)). More specifically, a t-test uses sample information to assess how plausible it is for the population means \(\mu_1\) and \(\mu_2\) to be equal. The test has two non-overlapping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population means, specifically the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a two sample t-test for two population means are:

  • Depending on our knowledge about the "no effect" situation, the t-test can be two-tailed, left-tailed or right-tailed

  • The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true

  • The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true

  • In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

The formula for a t-statistic for two population means (with two independent samples), with unknown population variances depends on whether the population variances are assumed to be equal or not. If the population variances are assumed to be unequal, then the formula is:

\[t = \frac{\bar X_1 - \bar X_2}{\sqrt{ \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} }}\]

On the other hand, if the population variances are assumed to be equal, then the formula is:

\[t = \frac{\bar X_1 - \bar X_2}{\sqrt{ \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}(\frac{1}{n_1}+\frac{1}{n_2}) } }\]

Normally, the way of knowing whether the population variances must be assumed to be equal or unequal is by using an F-test for equality of variances.

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