I'm struggling with the kind of statistics I should use for the following problem.

Some bridges in Amsterdam are built on wooden poles (in order for them to not sink due to the high level of the groundwater). I need to evaluate the strength of the wooden foundation on the basis of the strength of a small sample of wooden poles.

The following information is available:

- The bridge has 509 poles underneath

- We have a sample of 35 poles for which we know the stength

- The average strength of these 35 poles is 439 kN

- The standard deviation is 48 kN

- The minimal strength a pole should have is 185 kN

Now I would like to answer the following two questions:

1) What is the probability that all 509 poles have a minimal strength of 185 kN (and can I make such a statement with a sample consisting of 35 poles)?

2) How big should my sample actually be if I want to be able to state with a probabability of 95% that all 509 poles have a minimal strength of 185 kN?

My background is in high energy physics in which I'm used to work with PDFs (probability density functions) for very large samples. I'm new to problems like the above. Therefore I'm not certain which statistics to use.

Can I just use a normal distribution (constructed with the average and standard deviation of the sample of 35 poles) and integrate it from minus infinity to 185kN in order to get the probability asked in question 1? In that case I would not use the fact that the total population conists of 509 poles. Would this be the right way to go?

Or should I work with a student's t test and calculate the probability P and the accuracy Z? If so, could someone explain me how to proceed with the students t test for this specific problem?

I hope my questions are somewhat clear.

Any help is very much appreciated!