Confidence Intervals
A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. The width of the confidence interval gives us an idea of how uncertain we are about the unknown parameter.
Where:
- Point Estimate = Sample mean or proportion
- Critical Value = z* or t* value based on confidence level
- Standard Error = Standard deviation of the sampling distribution
Interval Type
Select the type of confidence interval you want to calculate:
Confidence Interval for Mean (Population Standard Deviation Known)
Use this when you know the population standard deviation. This uses the z-distribution.
Confidence Interval Results
Sample Mean (X̄)
Margin of Error
Confidence Interval
Sample Size (n)
Interpretation
Confidence Interval for Mean (Population Standard Deviation Unknown)
Use this when you don't know the population standard deviation. This uses the t-distribution.
Confidence Interval Results
Sample Mean (X̄)
Sample Standard Deviation (s)
Margin of Error
Confidence Interval
Interpretation
Sample Statistics
| Statistic | Value |
|---|---|
| Sample Size (n) | |
| Degrees of Freedom | |
| Standard Error of Mean |
Confidence Interval for Proportion
Use this to estimate a population proportion based on a sample proportion.
Confidence Interval Results
Sample Proportion (p̂)
Margin of Error
Confidence Interval
Interpretation
Calculation Details
| Statistic | Value |
|---|---|
| Number of Successes (x) | |
| Sample Size (n) | |
| Standard Error |
Confidence Interval Examples in Industrial Engineering
Quality Control Example
A quality engineer measures the diameter of 50 bolts and finds a sample mean of 10.2 mm with a known population standard deviation of 0.5 mm. They calculate a 95% confidence interval to estimate the true mean diameter of all bolts produced.
Process Improvement Example
An industrial engineer wants to estimate the proportion of defective items in a production process. From a sample of 200 items, 12 are found to be defective. They calculate a 99% confidence interval for the true proportion of defective items.
Time Study Example
A work measurement analyst times 15 cycles of a task and wants to estimate the true average time with 90% confidence. Since the population standard deviation is unknown, they use the sample standard deviation and the t-distribution.
Understanding Confidence Intervals
| Confidence Level | Meaning | Common z* Values |
|---|---|---|
| 90% | We are 90% confident that the interval contains the true parameter | 1.645 |
| 95% | We are 95% confident that the interval contains the true parameter | 1.960 |
| 99% | We are 99% confident that the interval contains the true parameter | 2.576 |