The two-sample t procedures Because we do not know the population standard deviations, we estimate them by Draw an SRS of size n, from a normal population the sample standard deviations from our with unknown mean u,, and draw an independent two samples. The result is the standard SRS of size n, from another normal population error, or estimated standard deviation of with unknown meanu. The confidence interval he difference in sample means: for u-u, given t x-元)土1+5 nn SE=./+ has confidence level at least C. here t is the Yn, n2 upper(1-C)/2 critical value for the t(k) distribution with k the smaller of n,-l and n-1 When we standardize the estimate by To test the hypothesis Ho: A=H2 dividing it by its standard error, the result compute the two-sample t statistic is the two-sample t statistic x SE The statistic t has the same interpretation as and use P-value or critical values for the t(k) distribution the true p-value or fixed any z or t statistic: it says how far x1-x2 is from o in standard deviation units significance level will always be equal to or less than the value calculated from t(k)27 53 • Because we do not know the population standard deviations, we estimate them by the sample standard deviations from our two samples. The result is the standard error, or estimated standard deviation, of the difference in sample means: 2 2 1 2 1 2 SE s s n n = + 54 • When we standardize the estimate by dividing it by its standard error, the result is the two-sample t statistic: 1 2 SE x x t − = The statistic t has the same interpretation as any z or t statistic: it says how far is from 0 in standard deviation units.1 2 x − x 28 55 The two-sample t procedures • Draw an SRS of size from a normal population with unknown mean , and draw an independent SRS of size from another normal population with unknown mean . The confidence interval for given by has confidence level at least C. Here t is the upper (1-C)/2 critical value for the t(k) distribution with k the smaller of and . 1 n 2 n μ1 μ2 μ1 2 − μ 2 2 1 2 1 2 1 2 ( ) s s xx t n n −± + 1 n −1 2 n −1 56 • To test the hypothesis , compute the two-sample t statistic 01 2 H : μ = μ 1 2 2 2 1 2 1 2 x x t s s n n − = + and use P-value or critical values for the t(k) distribution. The true P-value or fixed significance level will always be equal to or less than the value calculated from t(k)