Certainty Affairs I have a 3000 telephone number list. A computer software has found (automatically) that all the numbers are out of service. My boss mistrusts of that software, so he has asked me to pick a sample list in order to do the calls manually. That sample list must be created in a random way. What should be minimum size of the sample list necessary to justify the software? I'm not a mathematician but i heard something about the Chi-squared distribution.
Thanks in advance. 
PD: It seems homework, but it is not!!! This is a real life problem ToT.
 A: There is no simple answer aside from calling them all if you want to validate that they are all out of service.  If you want to validate that at least some percentage are out of service, you can do a sample.  Say you want to disprove that at least $10\%$ are still in service.  Then if you make $10$ calls and find all are out of service, you would say that if $10\%$ were in service I would have roughly a $1-0.9^{10} \approx 65\%$ chance of finding at least one working.  If you try $20$ calls, you would have $1-0.9^{20} \approx 88\%$ chance of finding at least one working.  You have a tradeoff between the percentage working, the confidence that the true value is below that, and the number of calls.
A: See if you can get information from some of the phone companies about how many numbers go in and out of service at a time and their current portion of active numbers over inactive numbers. If they're similar (statistically), you can pretend they're all from an imaginary typical company.
Imagine the process of dropping or gaining your phone number as a state machine. Initially, there's a portion $a=\text{active/numbers}$ of the numbers which are in service, and a portion $i=\text{inactive/numbers}=1-a$ whose are not. This would form the first branch of the probability tree, with an edge $\text{Start}\rightarrow \text{Active}$ labelled $a$ and likewise to Inactive by $i$. However, in a stochastic matrix model, you would use these as the coordinates of a vector $[a,i]$, called the initial state vector $S_0$.
Next, you have the change of state. The first pair of numbers to look at are the portion of numbers that go out of service $a'_1$ over some time period and the portion of numbers that remain in service $a'_2$ out of $a$. As a ratio of the numbers in service before they lost or kept numbers, it won't matter if the figures are out of date, and hence not matched to a portion of $a$. These form the 2nd branch of the probability tree extending from the node for active numbers. Likewise, $i_1, i_2$ are the portions of numbers that remain inactive or become reassigned, and complete the 2nd level of the probability tree.
Form a $2 \times 2$ matrix whose columns describe activity/inactivity probabilities for a state change, and rows the different starting states those probabilities are relevant to:
$P=\begin{pmatrix}a_1 & a_2 \\ i_1 & i_2\end{pmatrix}$
At this time you want to find the stationary probability vector for $S_0, P$, which is the eigenvector of $P$ which $S_0$ tends to as $P$ is applied repeatedly to it. In particular, it should give us a pair of probabilities, a stable version of $S_0$ whose coordinates have the same meaning, but for asymptotic likelihood of activity or inactivity, and hence something that better describes the long-term relevance of an outdated document from an imaginary company.
We'll call this vector $S'=[c_a,c_i]$, and define it to be the vector such that $S'P=S', c_a+c_i=1$. Hence,
$[c_a a_1+c_i i_1, c_a a_2 + c_i i_2]=[c_a, c_i],$ or 
$\begin{cases}
c_a (a_1-1)+c_i i_1 = 0, \\
c_a a_2 + c_i (i_2-1) = 0, \\
c_a+c_i = 1
\end{cases}$
