This calculator finds a two-sided z confidence interval for a population mean from the sample mean, the known population standard deviation σ, sample size and confidence level. It keeps decimal confidence levels and performs calculations before rounding the displayed results.
If you’re keen to delve deeper into the concept of confidence intervals, discover how they are calculated, or need the 95% confidence interval formula for z-scores, you’re in the right place.
Calculate a two-sided z confidence interval for a population mean using the known population standard deviation, σ. Observations must be independent, and the population or the sampling distribution of the mean must be approximately normal. If σ is unknown and you use a sample estimate s, a t interval is generally appropriate instead.
What is a confidence interval?
A confidence interval gives plausible values for a population parameter through a method with specified long-run coverage. A 95% method would capture the fixed true parameter in about 95% of intervals from repeated random samples under the same assumptions.
For this z interval, observations must be independent and the population or the sampling distribution of the mean must be approximately normal. The standard deviation entered must be the known population σ. If it is only a sample estimate s, use an appropriate t interval instead; this calculator does not compute t intervals.
This concept is fundamental in statistics because it allows us to make informed guesses about a population parameter with an acknowledged degree of uncertainty. Rather than providing a single estimate, confidence intervals reflect the variability in the data and give a range that is likely to include the true unknown parameter. This approach is crucial for scientific studies and data-driven decision-making where precision and reliability are essential.
How to calculate a confidence interval?
Suppose an independent sample has \(n=150\) and sample mean \(\bar{X}=4\), and the population standard deviation is known to be \(\sigma=0.7\). For a two-sided 95% z confidence interval, calculate the standard error and margin of error without rounding intermediate values.
- Start with the CI formula.
\( CI = \bar{x} \pm z \times \left(\frac{\sigma}{\sqrt{n}}\right) \) - Refer to a statistical table to find the Z(0.95)-score, which is the 97.5th quantile of the standard normal distribution, N(0,1). In this scenario, it equates to 1.96.
- Determine the standard error of the mean, calculated as \(\frac{\sigma}{\sqrt{n}}=\frac{0.7}{\sqrt{150}}\approx0.057\).
- Multiply the standard error by the z-score to find the margin of error: \(0.057\times1.96\approx0.112\).
- Add and subtract this margin of error from the mean to establish the confidence interval. Here, the interval ranges from 3.888 to 4.112.
That’s it. You’ve calculated the CI for \(n=150\), \(\bar{X}=4\), and a confidence level of 95%. Although it might seem like a lot, our Confidence Interval Calculator at MathBlog simplifies all these calculations for you, providing quick and accurate results.