SDC 3 Appendix: Background information and formulas for the p-values and confidence interval for the problem of comparing two means. Brief summary: The purpose of this technical appendix is to present the detailed formulas used to compute the p-values and the confidence interval for the problem of comparing two means. The derivation of the formulas relating pb and ph to p-values and the derivation of the confidence interval are also presented.. The model we assume is that Y11 …Y1n1 are independent and normally distributed with mean 1 and variance 12 and, independently, Y21 , Y2n are independent and normally distributed 2 2 with mean 2 and variance 2 . The problem is to make inferences about the difference in means 2 1 . The sample mean of sample i 1 2 is Y i ni1 ji1 Yij (presented in L62 and L93), and the n sample variance of sample i 1 2 is si2 (ni 1) 1 ji1 (Yij Y i )2 , (the sample standard n deviation si is presented in L63 and L94). The standard error is SE Y 2 Y 1 s12 n1 s22 n2 (presented in L64 and L95) and the approximate degrees of freedom is SE Y 2 Y 1 s12 n1 n1 1 s22 n2 n2 1 (presented in L65 and 4 2 2 L96). Under the assumed model, the sampling distribution of Y 2 Y 1 2 1 SE Y 2 Y 1 is approximated by the Student t distribution with degrees of freedom. In the equal variance case ( 12 22 ), the sampling distribution of Y 2 Y 1 2 1 SE Y 2 Y 1 where SE Y 2 Y 1 n 1 s n 2 1 1 2 1 s22 n1 n2 2 1 n1 1 n2 , is exactly the Student t distribution with n1 n2 2 degrees of freedom. Let G be the distribution function of the Student t distribution with degrees of freedom and let T denote a random variable with this distribution. The Student t distribution is symmetric so T and T have the same distribution. The one-sided p-value for testing the null hypothesis that 2 1 against the alternative that 2 1 is Pr T Y2 Y1 / SE Y2 Y1 2 1 Pr T Y2 Y1 / SE Y2 Y1 2 1 Pr T Y2 Y1 / SE Y2 Y1 2 1 G Y2 Y1 / SE Y2 Y1 ph given by [2]. Thus ph can be interpreted as the one-sided p-value for testing the null hypothesis that 2 1 against the alternative that 2 1 (i.e. the probability under the null hypothesis 2 1 that we observe a t-ratio greater than or equal to the observed value of the t-ratio Y 2 Y 1 / SE Y 2 Y 1 ). Similarly, pb can be interpreted as the one-sided p-value for testing the null hypothesis that 2 1 against the alternative that 2 1 . The derivation is Pr Tv Y2 Y1 / SE Y2 Y1 2 1 Pr Tv Y2 Y1 / SE Y2 Y1 2 1 Pr Tv Y2 Y1 / SE Y2 Y1 2 1 1 Gv Y2 Y1 / SE Y2 Y1 pb . The confidence interval [1] for 2 1 is obtained as 1 Pr t Y 2 Y 1 2 1 SE Y 2 Y 1 t Pr Y 2 Y 1 t SE Y 2 Y 1 2 1 Y 2 Y 1 t SE Y 2 Y 1 . The critical value t G 1 1 / 2 , where G 1 is the inverse of the distribution function of the Student t distribution.