How to compare two experimental samples I am conducting an experiment on a mechanical device. The theoy is that there is a function that maps an input force ($F_I$) to the otput ($F_O$), i.e. $F_O = f(F_I)$ and vice versa (function $f$ is linear). My goal is to test this theory in real-life conditions with an influence of numerous factors. Due to some limitations I can only measure some critical input force $F_{Icr}$ which corresponds to the device failure. Measuring it several times I obtain my first sample which I can convert (using the theory) to $F_{Ocr}$. Next I apply external force until failure and obtain my second sample of $F_{Ocr}$, but now it is measured at the output. Now I can somehow compare these two samples of the output force which are, in theory, should be the same.
The question is how to do it properly. I want to present my results and conclusion with some statistical background behind them. Sample size is around 10-20, since there is a lot of work to do applying forces frome different directions etc. Should I just use a simple t-test (my first samples of size 10 appear to be Gaussian if tested by Anderson-Darling test)? I am not even sure is these samples should be treated as dependent or not and need some advice on what tests should I use.
On the next stage I want to introduce some factors (stiffness in particular) and study how they affect my results. What would be the more reasonable option to analyse possible correlation for these small samples?
 A: $\newcommand{\si}{\sigma}$
Let $X_1,X_2,\dots$ be iid random variables (r.v.'s) representing the measurements of the input force. Let $Y_1=a+bX_1,Y_2=a+bX_2,\dots$ be iid r.v.'s representing the measurements of the output, for some real $a$ and $b$. We assume that $X_i\sim N(\mu,\si^2)$.  
Only the observed values $x_1,\dots,x_m$ of $X_1,\dots,X_m$ and $y_{m+1},\dots,y_{m+n}$ of $Y_{m+1},\dots,Y_{m+n}$ are known to us, for some sample sizes $m$ and $n$. The logarithm of the likehood function (that is, of the joint pdf) of $X_1,\dots,X_m,Y_{m+1},\dots,Y_{m+n}$ is given by the formula 
$$\ln L=c-\frac{m+n}2\,\ln(\si^2)-\frac n2\,\ln(b^2)
-\frac1{2\si^2}\,\sum_1^m(x_i-\mu)^2
-\frac1{2b^2\si^2}\,\sum_{m+1}^{m+n}(y_i-a-b\mu)^2,
$$
where $c:=\ln((2\pi)^{-m/2-n/2})$. We assume that the parameters $\mu,\si,a,b$ are all unknown, and we want to estimate them for this model, based on the data $x_1,\dots,x_m,y_{m+1},\dots,y_{m+n}$. 
Note that $L$ will not change if $b$ and $a$ are replaced by $-b$ and $a+2\mu$, respectively. Therefore, the sign of $b$ is not identifiable in this model and, at best, we can only estimate $b^2$. 
Let us now find the maximum likelihood estimators (MLEs) of $\mu,\si^2,a,b^2$. 
Taking the partial derivative of $\ln L$ in $a$, we see that the maximum of $L$ in $a$ occurs at 
\begin{equation*}
 a=\bar y-b\mu, \tag{1} 
\end{equation*}
where 
\begin{equation*}
 \bar y:=\frac1n\,\sum_{m+1}^{m+n}y_i,\quad \bar x:=\frac1m\,\sum_1^m x_i,  
\end{equation*}
so that we can write 
$$\ln L=c-\frac{m+n}2\,\ln(\si^2)-\frac n2\,\ln(b^2)
-\frac1{2\si^2}\,\sum_1^m(x_i-\mu)^2
-\frac{ns_y^2}{2b^2\si^2},  
$$
where 
\begin{equation*}
 s_y:=\sqrt{\frac1n\,\sum_{m+1}^{m+n}(y_i-\bar y)^2},\quad 
 s_x:=\sqrt{\frac1m\,\sum_1^m(x_i-\bar x)^2}. 
\end{equation*}
Now we see that the maximum of $L$ in $\mu$ occurs at 
\begin{equation*}
 \mu=\bar x,  \tag{2} 
\end{equation*}
so that we can write 
$$\ln L=c-\frac{m+n}2\,\ln(\si^2)-\frac n2\,\ln(b^2)
-\frac{ms_x^2}{2\si^2}
-\frac{ns_y^2}{2b^2\si^2}. 
$$
Now we see that the maximum of $L$ in $b^2$ occurs at 
\begin{equation*}
 b^2=s_y^2/\si^2,  \tag{3} 
\end{equation*}
so that we can write 
\begin{align*}
 \ln L&=c-\frac{m+n}2\,\ln(\si^2)-\frac n2\,\ln(s_y^2/\si^2)
-\frac{ms_x^2}{2\si^2}-n/2 \\  
&=c-\frac m2\,\ln(\si^2)-\frac n2\,\ln(s_y^2)
-\frac{ms_x^2}{2\si^2}-n/2. 
\end{align*}
Now we see that the maximum of $L$ in $\si^2$ occurs at 
\begin{equation*}
 \si^2=s_x^2.   \tag{4} 
\end{equation*}
Collecting (4)--(1), we conclude that the values of the MLEs of $\si^2,b,\mu,a$ are, respectively,
\begin{equation*}
 \hat\si^2:=s_x^2,\quad \hat b:=\pm s_y/s_x,  
\end{equation*}
\begin{equation*}
 \hat\mu:=\bar x,\quad 
 \hat a:=\bar y\mp\frac{s_y}{s_x}\hat x.  
\end{equation*}
