Suppose we have a sample $\{x_1, \ldots ,x_n\}$ obtained from i.i.d RV's $X_i\sim N$ (normal). Then in books on statistics, depending on the values obtained, they subtract a fixed quantity $L$ from every $x_i$, to get smaller values for computations, resulting in the sample $\{x_1 - L, \ldots, x_n -L\}$. The books states that this has no influence on statistical descisions based on the sample (why?).
Below I've attached a picture from a practical case (using SAS), where the sample consist of values $x_i \approx \pm 40$. Since the sample values $x_i$ are close to $40$ the book has decided to substract $40$ from all $x_i$. The variable diameter is the original values, while the variable diam_40 are the values with $40$ subtracted from them. Indeed the computed values seems pretty similar, however $t$ values are different also $Pr >$.
Why can we subtract from every sample value without getting different results "later on" ? How can I show my descisions will always be the same regardless of having subtracted or not ?
The $t$ values are calculated for two different hypotheses. The hypothesis test carried out is $$ \frac{\bar{x}-\mu^{H_0}}{\sqrt{\hat{V}(\bar{x})}}. $$ The null value is $\mu^{H_0}=0$ in both cases. But a null value of $0$ in the second case is equivalent to a null of $40$ in the first equation! Let $\bar{z}=\bar{x}-40$ be the transformed mean. The variance is unaffected. The first test in the output is $$ \frac{\bar{x}}{\sqrt{\hat{V}(\bar{x})}}. $$ and the second is $$ \frac{\bar{z}}{\sqrt{\hat{V}(\bar{z})}}=\frac{\bar{x}-40}{\sqrt{\hat{V}(\bar{x})}} $$ which is really the test in my first equation but with the null hypothesis $\mu=40$.