Section 7.2 Summary

• Significance tests and confidence intervals for the difference of the means ${\mu }_1$ and ${\mu }_2$ of two Normal populations are based on the difference ${\overline{x}}_1-{\overline{x}}_2$ of the sample means from two independent SRSs. Because of the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample sizes are large.

• When independent SRSs of sizes $n_1$ and $n_2$ are drawn from two Normal populations with parameters ${\mu }_1,{\sigma }_1$ and ${\mu }_2,{\sigma }_2$ the two-sample $z$ statistic

  $z={\displaystyle \frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{{\displaystyle \frac{{\sigma }_1^2}{n_1}}+{\displaystyle \frac{{\sigma }_2^2}{n_2}}}}}$

  has the $N\left(0,1\right)$ distribution.

• The two-sample $t$ statistic

  $t={\displaystyle \frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{{\displaystyle \frac{s_1^2}{n_1}}+{\displaystyle \frac{s_2^2}{n_2}}}}}$

  does not have a $t$ distribution. However, software can give accurate $P$-values and critical values using the Satterthwaite approximation.

• Conservative inference procedures for comparing ${\mu }_1$ and ${\mu }_2$ use the two-sample $t$ statistic and the $t\left(k\right)$ distribution with degrees of freedom $k$ equal to the smaller of $n_1-1$ and $n_2-1$. Use this method unless you are using software.

• An approximate level $C$ confidence interval for ${\mu }_1-{\mu }_2$ is given by

  $\left({\overline{x}}_1-{\overline{x}}_2\right)\pm t^{*}\sqrt{{\displaystyle \frac{s_1^2}{n_1}}+{\displaystyle \frac{s_2^2}{n_2}}}$

  Here, $t^{*}$ is the value for the $t\left(k\right)$ density curve with area $C$ between $-t^{*}$ and $t^{*}$, where $k$ either is found by the Satterthwaite approximation or is the smaller of $n_1-1$ and $n_2-1$. The margin of error is

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  $t^{*}\sqrt{{\displaystyle \frac{s_1^2}{n_1}}+{\displaystyle \frac{s_2^2}{n_2}}}$

• Significance tests for $H_0:{\mu }_1={\mu }_2$ are based on the two-sample $t$ statistic

  $t={\displaystyle \frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{{\displaystyle \frac{s_1^2}{n_1}}+{\displaystyle \frac{s_2^2}{n_2}}}}}$

  The $P$-value is approximated using the $t\left(k\right)$ distribution, where $k$ either is found by the Satterthwaite approximation or is the smaller of $n_1-1$ and $n_2-1$.

• The guidelines for practical use of two-sample $t$ procedures are similar to those for one-sample $t$ procedures. Equal sample sizes are recommended.

• If we can assume that the two populations have equal variances, pooled two-sample $t$ procedures can be used. These are based on the pooled estimator

  $s_p^2={\displaystyle \frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}}$

  of the unknown common variance and the $t\left(n_1+n_2-2\right)$ distribution.