Section 7.2 Summary

• Significance tests and confidence intervals for the difference of the means and of two Normal populations are based on the difference 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 and are drawn from two Normal populations with parameters and the two-sample statistic

  has the distribution.

• The two-sample statistic

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

• Conservative inference procedures for comparing and use the two-sample statistic and the distribution with degrees of freedom equal to the smaller of and . Use this method unless you are using software.

• An approximate level confidence interval for is given by

  Here, is the value for the density curve with area between and , where either is found by the Satterthwaite approximation or is the smaller of and . The margin of error is

  393

• Significance tests for are based on the two-sample statistic

  The -value is approximated using the distribution, where either is found by the Satterthwaite approximation or is the smaller of and .

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

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

  of the unknown common variance and the distribution.