# Margin of Error

See how to find the margin of error

(+ binomial confidence interval).

1 example and its solution.

## Binomial Confidence Interval

### Confidence Interval: Binomial Distribution to Normal Approximation

p̂ - Z√p̂q̂n ≤ p ≤ p̂ + Z√p̂q̂n

p̂: Probability of 'want' in the sample
(p̂ is read as [p hat].)

q̂: 1 - p̂

Probability of 'not want' in the sample

Z: Z-score

n: Sample size

In reality, it's hard to find the probability of 'want' in the population p.

(It takes so much time and effort.)

So, instead of finding the exact value of p,

we guess that p is in this confidence interval.

## Margin of Error

### Formula

(margin of error) = Z√p̂q̂n

### Example

A poll was taken of a random sample of 256 people.

64% of the people favored the candidate A.

For a 95% confidence level, find the margin of error.

(Assume P(-1.96 ≤ Z ≤ 1.96) = 0.95.)

Solution
64% of the people favored the candidate A.

For a 95% confidence level, find the margin of error.

(Assume P(-1.96 ≤ Z ≤ 1.96) = 0.95.)

n = 256

p̂ = 0.64

q̂ = 1 - 0.64

= 0.36

95% confidence level

→ Z = 1.96

(margin of error) = 1.96⋅√0.64⋅0.36256

= 1.96⋅√64⋅36256⋅100

= 1.96⋅√8

= 1.96⋅8⋅616⋅100 - [1]

= 1.96⋅62⋅100

= 1.96⋅3100

= 1.96⋅0.03

= 0.0588

= 5.88% - [2]

p̂ = 0.64

q̂ = 1 - 0.64

= 0.36

95% confidence level

→ Z = 1.96

(margin of error) = 1.96⋅√0.64⋅0.36256

= 1.96⋅√64⋅36256⋅100

^{2}= 1.96⋅√8

^{2}⋅6^{2}16^{2}⋅100^{2}= 1.96⋅8⋅616⋅100 - [1]

= 1.96⋅62⋅100

= 1.96⋅3100

= 1.96⋅0.03

= 0.0588

= 5.88% - [2]

[2]

This means for a 95% confidence level,

the actual favor rating p is in

(64 - 5.88)% ~ (64 + 5.88)%.

the actual favor rating p is in

(64 - 5.88)% ~ (64 + 5.88)%.

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