# Normal Approximation: Binomial

How to use the normal approximation to a binomial distribution: definition, 1 example, and its solution.

## Definition

### Definition

For a binomial distribution B(n, p),

if n is big,

then the data looks like

a normal distribution N(np, npq).

Using this property

is the normal approximation

to the binomial distribution.

So, when using the normal approximation

to a binomial distribution,

First change B(n, p) to N(np, npq).

Then standardize N(np, npq) to N(0, 1^{2}).

np: Expected value, mean

npq: (Standard Deviation)^{2}

## Example

### Example

### Solution

First, write the given condition as

B(n, p).

Getting a head of a coin

isn't affected by the previous trial.

So this is an independent event.

This is repeated.

So this is a binomial experiment.

n = 400

So write

B(400.

The probability of getting a head of a coin is

1/2.

So p = 1/2.

So write

1/2).

So the given binomial experiment is

B(400, 1/2).

B(400, 1/2)

p = 1/2

Then q = 1 - 1/2 = 1/2.

B(400, 1/2)

n = 400

p = 1/2

q = 1/2

Then, by the normal approximation,

this becomes

N(400⋅[1/2], 400⋅[1/2]⋅[1/2]).

400⋅[1/2] = 200

400⋅[1/2]⋅[1/2] = 100

100 = 10^{2}

So N(200, 10^{2}).

The mean (expected value) is 200.

The standard deviation is 10.

The probability of

getting a head 185 ~ 210 times

is P(185 ≤ X ≤ 210).

Find the z-score of 185.

N(200, 10^{2})

x = 200

σ = 10

Then Z = [185 - 200]/10.

185 - 200 = -15

-15/10 = -1.5

So the z-score of X = 185 is

Z = -1.5.

Find the z-score of 210.

N(200, 10^{2})

x = 200

σ = 10

Then Z = [210 - 200]/10.

210 - 200 = 10

10/10 = 1

So the z-score of X = 210 is

Z = 1.

X = 185 is Z = -1.5.

X = 210 is Z = 1.

So

P(185 ≤ X ≤ 200) = P(-1.5 ≤ Z ≤ 1).

Draw the normal distribution curve

like this.

Color the region

under the curve -1.5 ≤ Z ≤ 1.

The blue colored area is

P(-1.5 ≤ Z ≤ 0).

The left side and the right side are the same.

So P(-1.5 ≤ Z ≤ 0) = P(0 ≤ Z ≤ 1.5).

See the given z-score table.

P(0 ≤ Z ≤ 1.5) = 0.4332

So the blue colored area is

0.4332.

The green colored area is

P(0 ≤ Z ≤ 1).

See the given z-score table.

P(0 ≤ Z ≤ 1) = 0.3413

So the green colored area is

0.3413.

P(-1.5 ≤ Z ≤ 1) is the colored area

under the curve.

So

P(-1.5 ≤ Z ≤ 1)

= 0.4332 + 0.3413

= 0.7745.

So 0.7745, 77.45%, is the answer.