# Confidence Interval

See how to find the confidence interval.

2 examples and their solutions.

## Confidence Interval

### Formula

X - Z⋅σ√n ≤ m ≤ X + Z⋅σ√n

m: Mean of the population
Unknown, want to find

X: Mean of the sample

Z: Z-score

σ: Standard deviation of the sample

n: Sample size

In reality, it's hard to find the mean of the population m.

(It takes so much time and effort.)

So, instead of finding the exact value of m,

we guess that m is in this confidence interval.

95% Confidence Level

X - 1.96⋅σ√n ≤ m ≤ X + 1.96⋅σ√n

P(-1.96 ≤ Z ≤ 1.96) ≈ 0.95
99% Confidence Level

X - 2.58⋅σ√n ≤ m ≤ X + 2.58⋅σ√n

P(-2.58 ≤ Z ≤ 2.58) ≈ 0.99
### Example

Survey to find the mean of [high shcool students' time of eating lunch]: m minutes

Randomly picked 16 high school students

Mean: 15 minutes

Standard deviation: 4 minutes

Find the 95% confidence interval for m.

Solution
Randomly picked 16 high school students

Mean: 15 minutes

Standard deviation: 4 minutes

Find the 95% confidence interval for m.

n = 16

X = 15

σ = 4

15 - 1.96⋅4√16 ≤ m ≤ 15 + 1.96⋅4√16

15 - 1.96⋅44 ≤ m ≤ 15 + 1.96⋅44

15 - 1.96 ≤ m ≤ 15 + 1.96

14.04 ≤ m ≤ 16.96 - [1]

X = 15

σ = 4

15 - 1.96⋅4√16 ≤ m ≤ 15 + 1.96⋅4√16

15 - 1.96⋅44 ≤ m ≤ 15 + 1.96⋅44

15 - 1.96 ≤ m ≤ 15 + 1.96

14.04 ≤ m ≤ 16.96 - [1]

[1]

For a 95% confidence level,

m is in 14.04 minutes ~ 16.96 minutes.

m is in 14.04 minutes ~ 16.96 minutes.

Close

### Example

Survey to find the mean of [high shcool student's time of eating lunch]: m minutes

Randomly picked 16 high school students

Mean: 15 minutes

Standard deviation: 4 minutes

Find the 99% confidence interval for m.

Solution
Randomly picked 16 high school students

Mean: 15 minutes

Standard deviation: 4 minutes

Find the 99% confidence interval for m.

n = 16

X = 15

σ = 4

15 - 2.58⋅4√16 ≤ m ≤ 15 + 2.58⋅4√16

15 - 2.58⋅44 ≤ m ≤ 15 + 2.58⋅44

15 - 2.58 ≤ m ≤ 15 + 2.58

12.42 ≤ m ≤ 17.58 - [1]

X = 15

σ = 4

15 - 2.58⋅4√16 ≤ m ≤ 15 + 2.58⋅4√16

15 - 2.58⋅44 ≤ m ≤ 15 + 2.58⋅44

15 - 2.58 ≤ m ≤ 15 + 2.58

12.42 ≤ m ≤ 17.58 - [1]

[1]

For a 99% confidence level,

m is in 12.42 minutes ~ 17.58 minutes.

m is in 12.42 minutes ~ 17.58 minutes.

Close