5.6.1.2 - Unpooled Variances

5.6.1.2 - Unpooled Variances

When the assumption of equal variances is not valid, we need to use separate, or unpooled, variances. The mathematics and theory are complicated for this case and we intentionally leave out the details.

Hypothesis Tests for $$\mu_1− \mu_2$$: The Pooled t-test

Null:

$$H_0\colon\mu_1-\mu_2=0$$

Conditions:

We still have the following assumptions:

• The populations are independent
• Each population is either normal or the sample size is large

Test Statistic:

If the assumptions are satisfied, then

$$t^*=\dfrac{\bar{x}_1-\bar{x_2}-0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}$$

will have a t-distribution with degrees of freedom

$$df=\dfrac{(n_1-1)(n_2-1)}{(n_2-1)C^2+(1-C)^2(n_1-1)}$$

where $$C=\dfrac{\frac{s^2_1}{n_1}}{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}$$.

Note! This calculation for the exact degrees of freedom is cumbersome and is typically done by software. An alternate, conservative option to using the exact degrees of freedom calculation can be made by choosing the smaller of $$n_1-1$$ and $$n_2-1$$.
$$(1-\alpha)100\%$$ Confidence Interval for $$\mu_1-\mu_2$$ for Unpooled Variances
$$\bar{x}_1-\bar{x}_2\pm t_{\alpha/2} \sqrt{\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}}$$

Where $$t_{\alpha/2}$$ comes from the t-distribution using the degrees of freedom above.

Unpooled t-test

To perform a separate variance 2-sample, t-procedure use the same commands as for the pooled procedure EXCEPT we do NOT check box for 'Use Equal Variances.'

1. Choose Stat > Basic Statistics > 2-sample t
2. Select the Options box and enter the desired 'Confidence level,' 'Null hypothesis value' (again for our class this will be 0), and select the correct 'Alternative hypothesis' from the drop-down menu.
3. Choose OK.

For some examples, one can use both the pooled t-procedure and the separate variances (non-pooled) t-procedure and obtain results that are close to each other. However, when the sample standard deviations are very different from each other, and the sample sizes are different, the separate variances 2-sample t-procedure is more reliable.

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