﻿ Derivation of Confidence Interval

# Derivation of Confidence Interval

The central limit theorem states that, for samples of size n from a normal population, the distribution of sample means is normal with a mean equal to the mean of the population and a standard deviation equal to the standard deviation of the population divided by the square root of the sample size. (For suitably large sample sizes, the central limit theorem also applies to populations whose distributions are not normal.)

#### Central Limit Theorem

For samples of size n, the distribution of sample means

1. is normal.
2. has a mean of μ.
3. has a standard deviation of  .

where μ and σ represent the mean and the standard deviation of the population from which the sample came.

The distribution of sample means for samples of size n can be illustrated as follows:

For any given value of z* the probability that a sample mean lies within z* standard deviations of the mean can be calculated using ordinary left-tail probability tables. Let's call this probability C.

Notice, in particular, that this probability tells us something about the sample means but nothing about the population mean. Now let's consider the inequality:

Subtract μ from all three terms:

Subtract x-bar from all three terms:

Multiply all three terms by -1 remembering to reverse the inequalities:

Write the resulting inequality in an alternate form:

Substituting this result into our original probability we obtain:

In this form, C is called the confidence level and indicates how confident we are that the population mean lies within the indicated confidence interval. For example, if C = 0.95 then z* = 1.96. We say that we are 95% confident that the population mean lies within the interval: