90% Confidence Interval Calculator
Calculate 90% confidence intervals for population means with detailed statistical analysis
Calculate 90% Confidence Interval
The average value of your sample data
Population (σ) or sample (s) standard deviation
Number of observations in your sample
90% Confidence Interval Results
Calculation Details
Standard Error: SE = σ/√n = 0/0.00 = 0.0000
Critical Value: Z(0.90) = 1.645
Margin of Error: ME = 1.645 × 0.0000 = 0.0000
Confidence Interval: x̅ ± ME = 0 ± 0.0000
Interpretation
Example Calculation
Apple Box Weight Example
Scenario: A farmer wants to check apple box weights
Sample size (n): 170 boxes
Sample mean (x̅): 18.02 kg
Standard deviation (s): 1.5 kg
Confidence level: 90%
Step-by-Step Solution
1. Standard Error: SE = 1.5/√170 = 0.115
2. Critical Value: Z(0.90) = 1.645
3. Margin of Error: ME = 1.645 × 0.115 = 0.189
4. CI = 18.02 ± 0.189 = [17.83, 18.21] kg
Result: 90% confident the true mean weight is between 17.83 and 18.21 kg
Key Concepts
Confidence Interval
Range of values likely to contain the population parameter
Confidence Level
Probability that the interval contains the true parameter
Margin of Error
Maximum expected difference from the true value
When to Use Each Distribution
Z-Distribution
- • Known population σ
- • Large sample (n ≥ 30)
- • Normal population
t-Distribution
- • Unknown population σ
- • Small sample (n < 30)
- • Normal or nearly normal population
Understanding 90% Confidence Intervals
What is a Confidence Interval?
A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. A 90% confidence interval means we can be 90% confident that the true population mean lies within this range.
Key Components
- •Sample Mean (x̅): Average of your sample data
- •Standard Error (SE): Standard deviation divided by √n
- •Critical Value: Z-score (1.645) or t-score for 90% confidence
- •Margin of Error: Critical value × Standard error
Calculation Formula
Standard Error: SE = σ/√n
Margin of Error: ME = Critical Value × SE
Confidence Interval: x̅ ± ME
Lower Bound = x̅ - ME
Upper Bound = x̅ + ME
Critical Values
90% Confidence: Z = 1.645
95% Confidence: Z = 1.960
99% Confidence: Z = 2.576
Note: Use t-distribution when population standard deviation is unknown and sample size is small (n < 30).