Enter the confidence level, sample size, sample mean, and standard deviation to find the confidence interval.
How to Calculate Confidence Intervals
A confidence interval is a range of values that is likely to contain an unknown population parameter. It is commonly used in statistics to estimate the true value of a population mean.
Confidence Interval Formula
The formula for calculating a confidence interval for a population mean is:
\[ \text{CI} = \bar{x} \pm z \cdot \frac{s}{\sqrt{n}} \]
Where:
\( \bar{x} \) is the sample mean
\( z \) is the z-score for the chosen confidence level
\( s \) is the sample standard deviation
\( n \) is the sample size
Calculation Steps
Choose a confidence level (usually 90%, 95%, or 99%)
Determine the z-score for the chosen confidence level
Calculate the margin of error: \( z \cdot \frac{s}{\sqrt{n}} \)
Calculate the lower bound: \( \bar{x} - \text{margin of error} \)
Calculate the upper bound: \( \bar{x} + \text{margin of error} \)
Example Calculation
Let's calculate a 95% confidence interval for a sample with the following data:
Sample mean (\( \bar{x} \)) = 25
Sample standard deviation (\( s \)) = 2
Sample size (\( n \)) = 30
For a 95% confidence level, the z-score is 1.96
Calculate the margin of error:
\[ 1.96 \cdot \frac{2}{\sqrt{30}} \approx 0.7155 \]
Therefore, we can be 95% confident that the true population mean falls between 24.2845 and 25.7155.
Visual Representation
Here's a visual representation of a confidence interval:
This diagram illustrates a 95% confidence interval. The blue curve represents the normal distribution, the red line shows the interval, and the center point represents the sample mean.