according to the probability theory,
suppose that we calculate the experimental probability of students who prefer mathematics and it was .70%from a sample of 20 students (14/20), is that correct to use these percentage (70% to calculate the probability of prefer mathematics in case of applying the same survey on a sample of 200 students?
14/20 is a proportion or percentage. It's a relative frequency, not a probability. One may (!) use the relative frequeny as an estimate for the probability, that a student sampled under similar conditions will prefer mathematics (and this also only in the case that there is a countable set of possible alternatives).
Yes, if you have a sample of 20 and find that 14 of 20 students prefer mathematics and want to guess at the number of students that will prefer mathematics in a similar survey of 200 then multiply 200 by 0.70. Just don't be surprised if the results from the sample of 200 are quite different from the initial sample of 20.
In a sample of 20 my choices are 0%, 5%, 10%, ...95%, or 100%. What will my answer be if the true probability is 67.5%? In a sample size of 20 is it possible to accuratly estimate a probability of 67.5?% How might this be done keeping the sample size fixed at exactly 20?
14/20 is a proportion or percentage. It's a relative frequency, not a probability. One may (!) use the relative frequeny as an estimate for the probability, that a student sampled under similar conditions will prefer mathematics (and this also only in the case that there is a countable set of possible alternatives).
In addition to Jochen's point that this is an estimate of a probability, it also has an uncertainty bound that is extremely large. It is a point estimate. In R
> binconf(14,20)
PointEst Lower Upper
0.7 0.4810272 0.8545228
so your choice of any particular (even the most likely) point estimate (if you need a point estimate) should depend on the costs associated with having an estimate that is too low or too high. You might want to choose .6 depending on how safe you want to be in your n=200 sample predictions. For example, if I know the 99.99% estimate bounds for time for rice to boil are ten minutes plus/minus 2 minutes, I am going to be checking after 8 minutes because burnt rice is a pain to clean.
Samah Elbehary,
From a probability point of view, you have two random experiments with different sample and probability spaces. Then you can't use the results in one of them in the other.
From a statistical point of view, I agree with the answers given by Jochen and Daniel.
Dear Samah Elbehary,
In order for the answer to be "Yes", the sample should be large enough so that the central limit theorem can be applied. Since n=20 is fairly small, the proportion 14/20 is an approximate value to the probability of students who prefer mathematics. In any way, if your population is the 200 students in your question, then the exact proportion of students who prefer mathematics can be computed by just surveying these 200 students again, provided their decisions are independent.
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