Most of statistics textbooks approximate the binomial probabilities by the normal probabilities when testing binomial hypothesis especially when the sample size n is large enough.
Now, since we have CAS programs like SPSS and Minitab, which can compute the exact probabilities, do we really need to use the normal approximation of the binomial probabilities to compute, for example, the probability of committing a type I error?
Can't we just use Minitab, for example, to compute the exact binomial probabilities without using the Central Limit Theorem and the standard normal tables?
Thanks.
Normal approximation is an approximation only. I think this is not the more correct to do but it is a possibility when there is not other at hand.
You did not mention your research question? In much of our experimental data collection- we have assumptions about our particular data - this is a result of choices we have made.. In your case to collect the success or failure from a number of binomial trials etc. So in your design you may be collecting binomial data (0/1) which we often describe statistically as "success" and "failure". So maybe you have 50 to 100 trials (eg data from 50 to 100 subjects) . Another assumption is that all your data from these are independent.
So the test you seem to be interested in is a binomial, and here you are planning to use a computer to get answer.. One way we think of a test is to propose the null and the alternate hypothesis. If you use the binomial approximation, it is because your want an estimate the evidence to help answer the question. So the exact probability is not that important - you should set up your hypothesis before you do the test and decide what probability (say 0
It is true that the advent of the computer has made it unnecessary to use normal approximation to the binomial when the exact probabilities can be computed by computer. The teaching of the approximation is still necessary in schools. The approximate value might be as good as the exact value. It is still useful to know the approximation because only a range of values may be necessary rather than the exact value.
It is an APPROXIMATION when sample size is large (and CLT is applied). For EXACT probability use the Binomial.
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