The **Confidence Interval Calculator** calculates both the lower and upper bounds of the interval in which the estimate of a population mean (μ) is likely to fall. By inputting some key parameters and selecting a desired level of confidence, the calculator generates a range of values within which the true population parameter is estimated to lie

To use it you need the following input:

The mean of the sample, x

The size of the sample, n

The Standard Deviation of the sample, s

The confidence level (90%, 95%, or 99%), indicating "how sure" you want to be in estimating the true parameter in the original population (μ).

The bounds of the interval are obtained through the following formula:

Upper bound = x + z * s / sqrt(n) Lower bound = x - z * s / sqrt(n) where z represents the z-score corresponding to the confidence level chosen, i.e. the value of a normal distribution N(0,1) with mean 0 and standard deviation 1 that leaves to its right and left tails a probability of 10%, 5%, and 1%.

The Confidence Interval is a key concept in statistical inference, a branch of statistics that aims at drawing insights into a population based on randomly extracted samples. Statistical inference is crucial in making informed business decisions, as it allows analysts to make predictions and draw conclusions based on limited data.

At the theoretical foundation of statistical inference is the **Central Limit Theorem (CLT) **which states that as the sample size increases, the distribution of the sample means approaches a normal distribution with mean μ, regardless of the underlying distribution of the population from which the samples are drawn.

In other words, if a large enough sample size (n >30) is taken from a population, the sample means will be normally distributed, regardless of the shape of the original population distribution.