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 student's 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