Question

I have an original data set and 4 sets of transformed data to which I applied a two-sample pooled variance t-procedure in R. I am uncertain about to how to interpret the confidence intervals of the transformed data since when I back-transform the endpoints by undoing the original operation, I get values that don't seem anywhere near correct. Each of the transformations that was applied has been added as a suffix to the vector name containing the transformed data (in order: original data, log base e, squared root, cubed root, reciprocal).  

> t.test(X, Y, var.equal = TRUE)

Two Sample t-test

data: X and Y
t = 6.8315, df = 115, p-value = 4.224e-10
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
10381.71 18860.54
sample estimates:
mean of x mean of y
48001.52 33380.39

> t.test(X-LOG, Y-LOG, var.equal = TRUE)

Two Sample t-test

data: X-LOG Y-LOG
t = 6.893, df = 115, p-value = 3.112e-10
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
0.3120417 0.5636984
sample estimates:
mean of x mean of y
10.75884 10.32097

> t.test(X-SQRT, Y-SQRT, var.equal = TRUE)

Two Sample t-test

data: X-SQRT and Y-SQRT
t = 6.9433, df = 115, p-value = 2.423e-10
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
28.11332 50.55655
sample estimates:
mean of x mean of y
217.9947 178.6598

> t.test(X-CUBE, Y-CUBE, var.equal = TRUE)

Two Sample t-test

data: X-CUBE and Y-CUBE
t = 6.9446, df = 115, p-value = 2.407e-10
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
3.308980 5.949869
sample estimates:
mean of x mean of y
36.18075 31.55133

> t.test(X-RECIPROCAL, Y-RECIPROCAL, var.equal = TRUE)

Two Sample t-test

data: X-RECIPROCAL and Y-RECIPROCAL
t = 6.3703, df = 115, p-value = 4.029e-09
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
1.040067e-05 1.978757e-05
sample estimates:
mean of x mean of y
3.679029e-05 2.169616e-05

Answer 1