Probablity question with unknown distribution function. 
A manufacturer produces washers with a thickness $d$ that has a mean value of 1.0 mm and a standard deviation of 0.1 mm. If $N = 25$ washers are selected at random and stacked on top of one another, determine the probability that the height of the stack will be between 24 mm and 26 mm assuming that $d$ has an unknown distribution function.

Can I just use the fact that each washer is iid and thus use the $E[Nd] ~$ Normal($1.0N,0.1N$)?
That is the only way I could think of solving this problem, but I am uncertain because the washers thickness is not normal. 

Please allow me to modify your question in terms of just the variables. In other words, let the number of washers selected be $n=25$. Since $n$ is sufficiently large, we can apply Central Limit Theorem. The distribution of the sample follows a normal distribution. Give this distribution a mean of $\mu_\bar{x}$ and a SD of $\sigma_\bar{x}$. Here's how to calculate the 2 parameters of this new normal distribution:
$$
\begin{align*}
\mu_\bar{x} &= \frac{\sum x_1 + x_2 + ... + x_{25}}{25} \\
&=\frac{25 \times1.0}{25}\\
&=1.0
\end{align*}
$$
Now, for the $\sigma$, it is a little different
$$
\begin{align*}
\sigma^2_\bar{x} &= \frac{\sigma_1}{25} \\
&=\frac{0.1}{25}\\
&=0.004
\end{align*}
$$
