Confidence Interval Calculator with X O N N

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. This calculator helps you compute confidence intervals for a population mean (μ) when the population standard deviation (σ) is known, using the sample mean (X̄) and sample size (n).

What is a Confidence Interval?

A confidence interval provides an estimated range of values which is likely to contain an unknown population parameter. The most common confidence intervals are for the population mean, and they are typically calculated using the sample mean and standard deviation.

When calculating a confidence interval for a population mean, you need to know or estimate the population standard deviation. If the population standard deviation is known, you can use the z-distribution to calculate the confidence interval. If the population standard deviation is unknown, you can use the t-distribution.

This calculator assumes you know the population standard deviation (σ). If σ is unknown, use our Confidence Interval Calculator with t-distribution instead.

How to Use This Calculator

Enter the sample mean (X̄) - the average of your sample data.
Enter the sample size (n) - the number of observations in your sample.
Enter the population standard deviation (σ) - the standard deviation of the entire population.
Select the confidence level (typically 90%, 95%, or 99%).
Click "Calculate" to compute the confidence interval.

The calculator will display the confidence interval and provide an interpretation of the result.

Formula and Calculation

The formula for calculating a confidence interval for a population mean when the population standard deviation is known is:

Confidence Interval = X̄ ± (z × (σ/√n))

Where:

X̄ = sample mean
z = z-score corresponding to the selected confidence level
σ = population standard deviation
n = sample size

The z-score is derived from the standard normal distribution and corresponds to the selected confidence level. For example:

90% confidence level: z = 1.645
95% confidence level: z = 1.960
99% confidence level: z = 2.576

Interpreting Results

The confidence interval provides a range of values which is likely to contain the population mean. For example, if you calculate a 95% confidence interval of [45, 55], you can be 95% confident that the true population mean falls between 45 and 55.

Confidence intervals are useful for understanding the precision of your estimate. A narrower confidence interval indicates a more precise estimate, while a wider interval indicates more uncertainty.

Worked Example

Suppose you have a sample of 30 students with an average test score of 75 (X̄ = 75). The population standard deviation (σ) is known to be 10. You want to calculate a 95% confidence interval for the population mean test score.

X̄ = 75
n = 30
σ = 10
Confidence level = 95% (z = 1.960)

Using the formula:

Confidence Interval = 75 ± (1.960 × (10/√30))

Margin of Error = 1.960 × (10/5.477) ≈ 3.62

Lower Bound = 75 - 3.62 ≈ 71.38

Upper Bound = 75 + 3.62 ≈ 78.62

The 95% confidence interval for the population mean test score is approximately [71.38, 78.62]. This means we are 95% confident that the true population mean test score falls between 71.38 and 78.62.

FAQ

What is the difference between a confidence interval and a confidence level?

A confidence level is the percentage that the interval will contain the true population parameter (e.g., 95%). A confidence interval is the range of values calculated from the sample data that is likely to contain the population parameter.

How do I know if my sample size is large enough?

The sample size should be large enough to ensure the sample mean is normally distributed. A common rule of thumb is that the sample size should be at least 30, but this can vary depending on the population distribution.

What if the population standard deviation is unknown?

If the population standard deviation is unknown, you should use the t-distribution instead of the z-distribution. Our Confidence Interval Calculator with t-distribution is designed for this scenario.

Can I use this calculator for small sample sizes?

Yes, but the results may be less reliable. For small sample sizes, it's important to ensure that the sample is representative of the population and that the data is normally distributed.