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Notes › EENG 3421: Probability and Statistics for Engineers Lecture 27

Inference Tests for Two Means

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Independent Vs Dependent Samples

Definition 27.1 (Independent Versus Dependent Samples).

Two samples drawn from two populations are independent if the selection of one sample does not affect the selection of the second sample. Otherwise, the samples are dependent

Setup and Notation

Sampling Distribution of $\bar x_1 - \bar x_2$

Theorem 27.2 (Sampling Distribution of $\bar x_1 - \bar x_2$).

If

  • samples are independent
  • $\sigma_1,\sigma_2$ are known
  • samples are large or populations are normal then $\bar x_1 - \bar x_2$ is approximately normal with
$$ \mu_{\bar x_1-\bar x_2} = \mu_1 - \mu_2 $$$$ \sigma_{\bar x_1-\bar x_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} $$

Confidence Interval for $\mu_1 - \mu_2$

Theorem 27.3 (Confidence Interval for $\mu_1 - \mu_2$).

Using the normal model, the $(1-\alpha)100\%$ confidence interval for $\mu_1-\mu_2$ is

$$ (\bar x_1 - \bar x_2) \pm z\,\sigma_{\bar x_1-\bar x_2} $$

where

$$ \sigma_{\bar x_1-\bar x_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} $$

References

Sources

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