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Okay. So, many sources state different conditions for approximating binomial using normal. Variations I’ve seen are as follows. Assuming that we have Bin(n,p):

1. np and n(1-p) are both large (say, 20 or bigger)

2. large n, p close to 0.5

3. large n, np>9(1-p)

My questions:

1. Are all 3 of these valid? Justify your answer.

2. Is there a set of conditions which can summarise all 3 given above (if they are valid)?

Note: Before any of you say “First, what is your opinion about this?”, I personally don’t know much about these distributions since I’m studying stats at a basic level. I know quite a few distributions and a bit about pdf’s, cdf’s, mgf’s but not really that much more

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The usual conditions are that $n\geq 50$ and both $np$ and $n(1-p)>5$.

Note that if $p$ is close to $0.5$ your third condition is inconsistent with your first condition.

The Berry-Esseen theorem says that the uniform CDF error for approximating a sum of $n$ iid variables with 3 moments is bounded by $\frac{C \rho}{\sigma^2 \sqrt{n}}$, where $\rho=E[|X_i-m|^3]$ and $C$ is a universal constant. We do not exactly know the best possible value of $C$ but we do know that $0.4<C<0.5$. For the binomial distribution this gives a bound of

$$\frac{1}{2} \frac{p^3(1-p)+p(1-p)^3}{p^{3/2} (1-p)^{3/2} n^{1/2}}.$$

This bound is not tight, as can be checked by some direct calculation, but it is not bad (it is more pessimistic than the true error by a factor of maybe 4 or so). Cancelling as is possible, you get a bound of

$$\frac{1}{2} \frac{p^2+q^2}{(npq)^{1/2}}$$

where $q=1-p$. The numerator there is between $1$ and $1/2$, so we can intuitively understand this bound

$$\frac{C}{(npq)^{1/2}}$$

where what I have said so far tells us only that $C$ can be chosen to be smaller than $1/2$, though in actuality for the binomial distribution it can be chosen to be a fair bit smaller than that.

So if, say, you want a uniform error of at most $0.01$ then it will be enough (by the above analysis) to have $n>\frac{10000}{16 pq}$. In actuality the cutoff is probably more like $n>\frac{100}{pq}$.

First, versions of all three of your conditions can be useful. Practically speaking, the main goal is to keep the approximating normal distribution from putting much of its probability below $0$ or above $n.$

This idea gives rise to your #3 as follows: To keep most of the normal probability above $0$, you want normal parameters $\mu$ and

$\sigma$ to satisfy

$$ 0 < \mu – 3\sigma = np – 3\sqrt{np(1-p)}$$

Which implies

$ np > 3\sqrt{np(1-p)}$ or $np/(1-p) > 9.$ Some authors are happy enough

with $np/(1-p) > 5.$ In the best case, you can expect only about two-place accuracy when approximating binomial probabilities by normal ones.

Rhe stricter rule usually makes the approximation a little better. [A similar argument to ensure $n > \mu – 3\sigma$ gives the same kind

of bound: $n(1-p)/p > 9$ (or 5).]

The rule that the smaller of $np$ and $n(1-p)$ should exceed some number

is just a simplified version of the above where the denominator in the previous argument is set to 1. I have seen suggested bound as small as 5, but 20 will

usually give better results.

The rule that $p \approx 1/2$ just recognizes that the approximating normal

distribution is symmetrical, and the binomial is also symmetrical when $p=1/2.$

Saying that $n$ should be ‘large’ is always a safe suggestion.

Three further comments on normal approximation to the binomial are also relevant.

(a) It is important to use the continuity correction unless $n$ is several hundred.

(b) A Poisson approximation is often better than a normal approximation,

especially if $n$ is large and $p$ is small.

(c) Nowadays, modern software (R, SAS, Minitab, MatLab, etc.) provides the opportunity to get *exact* binomial probabilities. In practical situations

the normal approximation is usually avoided.

[But the approximation persists

in theoretical probability texts: It is a nice

illustration of the Central Limit Theorem. It makes doing

exercises possible by using the (still ubiquitous) printed table of the normal

CDF in the ‘back of the book’.]

**Some illustrative examples** using R statistical software.

*Breaks all the rules except $p = .5$, yet the normal approximation
happens to work quite well.*

Let $X \sim Binom(3, .5).$ Find $P(X = 2) = P(1.5 < X < 2.5).$

```
dbinom(2, 3, .5)
## 0.375 # exact
mu = 3*.5; sg = sqrt(3*.5*.5); diff(pnorm(c(1.5, 2.5), mu, sg))
## 0.3758935 # norm aprx
```

*Large $n$ and small $p$. Satisfies all rules, but normal approximation is
relatively poor and Poisson approximation is better.*

Let $X \sim Binom(1000, .03)$ Find $P(X \le 2)$.

```
pbinom(20, 1000, .03)
## 0.03328078 # exact via binom CDF
sum(dbinom(0:20, 1000, .03))
## 0.03328078 # exact via binom PDF
mu = 1000*.03; sg = sqrt(1000*.03 *.97); pnorm(20.5, mu, sg)
## 0.03911311 # relatively poor normal aprx
ppois(20, mu)
## 0.03528462 # better Poisson aprx
```

*Rules satisfied and normal approximation gives 2-place accuracy.*

Let $X \sim Binom(100, .3).$ Find $P(24 < X \le 35) = P(24.5 < X < 35.5).$

```
diff(pbinom(c(24, 35), 100, .3))
## 0.7703512 # exact
sum(dbinom(25:35, 100, .3))
## 0.7703512 # exact
mu = 100*.3; sg=sqrt(100*.3*.7); diff(pnorm(c(24.5, 35.5), mu, sg))
## 0.7699377 # relatively good normal aprx
diff(ppois(c(24, 35), mu))
## 0.6853745 # inappropriate Poisson 'aprx'
```

The figures below show blue bars for binomial probabilities, smooth curves for

normal densities, small circles for Poisson probabilities, and vertical dotted lines to bound approximating areas. Red indicates a relatively poor fit.

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