# 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.

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