t-tests & confidence limits

# input data y1=c(45,87,123,120,70) y2=c(51,71,42,37,51,78,51,49,56,47,58)

# Uneq var t-test of diff between means n1=length(y1) # assume no missing points n2=length(y2) v1=var(y1) # assume no missing points v2=var(y2) D=0 # true difference assumed by H0 d=mean(y1)-mean(y2) # observed difference # estimate standard error of d se=sqrt(v1/n1+v2/n2) t=(d-D)/se # observed t-statistic # find the corrected degrees of freedom df=(v1/n1+v2/n2)^2/((v1/n1)^2/(n1-1)+(v2/n2)^2/(n2-1)) p=pt(t,df) # test t p=.5-abs(p-.5) # when p > 0.5 then p = 1-p p*2 # 2-tailed P-value (method 1) t # observed t-value # corrected critical value (method 2) (qt(0.975,n1-1)*v1/n1+qt(0.975,n2-1)*v2/n2)/(v1/n1+v2/n2) d # observed difference between means d+qt(0.05/2,df)*se # lower 95% limit d-qt(0.05/2,df)*se # upper 95% limit

> # Uneq var t-test of diff between means > p*2 # 2-tailed P-value (method 1) [1] 0.07533183 > t # observed t-value [1] 2.306873 > # corrected critical value (method 2) > (qt(0.975,n1-1)*v1/n1+qt(0.975,n2-1)*v2/n2)/(v1/n1+v2/n2) [1] 2.746209 > > d # observed difference between means [1] 35.27273 > d+qt(0.05/2,df)*se # lower 95% limit [1] -5.462216 > d-qt(0.05/2,df)*se # upper 95% limit [1] 76.00767