The Confidence Interval Calculator for two samples helps you determine the range within which the true difference between two population means lies, based on your sample data. This is useful in comparing different groups to see if their means are significantly different from each other.
Formula of Confidence Interval Calculator Two Sample
To calculate the confidence interval for the difference between two sample means, use the following formulas:
If population variances are known:
CI = (X1 - X2) ± Z * sqrt((σ1^2 / n1) + (σ2^2 / n2))
If population variances are unknown, use sample variances:
CI = (X1 - X2) ± t * sqrt((s1^2 / n1) + (s2^2 / n2))
Degrees of freedom for the t-distribution can be calculated as:
df = ((s1^2 / n1) + (s2^2 / n2))^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
Where:
- X1 and X2 are the sample means
- Z is the Z-value for the desired confidence level
- t is the t-value for the desired confidence level and degrees of freedom
- σ1^2 and σ2^2 are the population variances
- s1^2 and s2^2 are the sample variances
- n1 and n2 are the sample sizes
Common Terms and Calculations
Below is a table of common terms and their corresponding Z-values for various confidence levels:
| Confidence Level | Z-value |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Example of Confidence Interval Calculator Two Sample
Consider two samples with the following data:
Sample 1: mean (X1) = 50, variance (s1^2) = 25, sample size (n1) = 30
Sample 2: mean (X2) = 45, variance (s2^2) = 20, sample size (n2) = 35
Assume we want a 95% confidence interval and the population variances are unknown. The t-value for 95% confidence level with degrees of freedom calculated is approximately 2.042.
CI = (50 - 45) ± 2.042 * sqrt((25 / 30) + (20 / 35))
CI = 5 ± 2.042 * 1.185 = 5 ± 2.419
Thus, the confidence interval is approximately (2.581, 7.419).
Most Common FAQs
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval.
If you have a 95% confidence interval, you can be 95% confident that the true population mean difference lies within the interval range. It does not mean that there is a 95% probability that the true mean difference is within the interval.
The t-distribution is used instead of the normal distribution when the population variances are unknown and the sample size is small. It accounts for the additional variability introduced by estimating the population standard deviation from the sample.