Question

1) Please explain binomial approximation.
2) How can it be used in calculating population size?
3) Please provide an example.

Answer 1

Answer :

Binomial approximation :-

On the off chance that you are working from a vast measurable example, tackling issues utilizing the binomial dispersion may appear to be overwhelming. Be that as it may, there's really a simple method to surmised the binomial dissemination, as appeared in this article.

Here's a precedent: assume you flip a reasonable coin multiple times and you let X break even with the quantity of heads. What's the likelihood that X is more prominent than 60?

In a circumstance like this where n is vast, the estimations can get inconvenient and the binomial table comes up short on numbers. So if there's no innovation accessible (like when taking a test), what would you be able to do to locate a binomial likelihood? Turns out, if n is sufficiently extensive, you can utilize the typical appropriation to locate a nearby surmised answer with much less work.

In any case, what do we mean by n being "sufficiently extensive"? To decide if n is sufficiently vast to utilize what analysts call the typical guess to the binomial, both of the accompanying conditions must hold:

To locate the ordinary estimate to the binomial dispersion when n is huge, utilize the accompanying advances:

Confirm whether n is sufficiently extensive to utilize the ordinary estimation by checking the two suitable conditions.

For the above coin-flipping question, the conditions are met in light of the fact that n ∗ p = 100 ∗ 0.50 = 50, and n ∗ (1 – p) = 100 ∗ (1 – 0.50) = 50, the two of which are something like 10. So proceed with the typical estimation.

Extending the entire thing utilizing Binomial Theorem gives you a correct esteem. Not an estimation.

To get an estimation you can think about a couple of terms from the development.

For example, for "little" x, 1+nx is a "sensible" estimate for (1+x)n.

Notice this compares to picking the initial two terms from the binomial hypothesis development (1+x)n=1+(n1) x+(n2) x2+⋯+xn.

Expanding the whole thing using Binomial Theorem gives you an exact value. Not an approximation.

To get an approximation you can consider a few terms from the expansion.

For instance, for "small" xx, 1+nx1+nx is a "reasonable" approximation for (1+x)n(1+x)n.

Notice that this corresponds to picking the first two terms from the binomial theorem expansion (1+x)n=1+(n1) x+(n2) x2+⋯+xn(1+x)n=1+(n1) x+(n2) x2+⋯+xn.

For example :-

1.00079≈1+9×0.0007=1.0063 which concurs with 1.00079=1.0063176688422737867054812736724 upto 4 decimal spots.

Contingent upon how exact you need it, you could think about more terms from the binomial development.

This depends on the way that for little x, as the power r of x gets bigger, the term xr turns out to be little very quick.