# 5.6.1.1 - Pooled Variances

5.6.1.1 - Pooled Variances## Hypothesis Tests for \(\mu_1− \mu_2\): The Pooled t-test

Now let's consider the hypothesis test for the mean differences with pooled variances.

**Null:**

\(H_0\colon\mu_1-\mu_2=0\)

**Conditions:**

The assumptions/conditions are:

- The populations are independent
- The population variances are equal
- Each population is either normal or the sample size is large

**Test Statistic:**

The test statistic is...

\(t^*=\dfrac{\bar{x}_1-\bar{x}_2-0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\)

And \(t^*\) follows a t-distribution with degrees of freedom equal to \(df=n_1+n_2-2\).

The p-value, critical value, and conclusion are found similar to what we have done before.