Hi!

I've been wrestling with this the whole day, statistics is really not my forte.

My problem is this:

In a population of 4 260, 30.19 % are positive to a certain product.

If I take a sample of 100, what is the probability that the proportion of people, in the sample taken, being positive to the product deviates max +/- 5 % from the proportion I got from the whole population (30.19 %).

So far I have gotten this answer on formula structure:

Pr{100(30.19%-5%) < X < 100(30.19%+5%) }

I've also gotten that X ~ Binomial(100, 32.19%)

However I have no idea how to use this. I need to be able to type a formula for the problem, calculate it and give the answers to it. We use SPSS as a program but from the question asked I believe I should be able to show the full calculations "by hand".

I've tried with the binomial formula but my calculator can't handle 100!/(0.2519!(100-0.2519)!) ... Anyway the way I tried to interpret the answer I've gotten is to type:

100!/(0.2519!(100-0.2519)!) * 0.3019^^{0.2519}(1-0.6981)^^{100-0.2519}thereby trying to calculate the probability of the sample proportion being max -5 % below.

I was then going to do the same for sample proportion being +5 % above and then just construct a Y<0.3019<X type interval as answer to the question. Am I way out in the blue here? Should I be using like z-tables or something instead?

What is the name of the formula for this? My book mentions Binomial distribution (discrete distributions), Probabilities in a uniform distribution (continuous distribution), "Using the normal curve to approximate binomial distribution". All of these have varying formula to use and I can't really find one where I can calculate an interval of a deviation from population proportion %, being +/- 5 % in a sample of 100.

As an edit:

How do I calculate how big of a sample I need to take if there is a requirement that the Standard Deviation to the estimated proportion is at the most 2.5 %. If I use the proportion of people being positive to the product in the population (30.19 %)?

Really grateful for any help!