4.5.2 - Derivation of the Confidence Interval

To calculate the confidence interval, we need to know how to find the z-multiplier. So where does this $$z_{\alpha}$$ come from?

The confidence interval can be derived from the following fact:

\begin{align} P\left(\left|\frac{\hat{p}-p}{\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}}\right|\le z_{\alpha/2}\right)=1-\alpha \\ P\left(-z_{\alpha/2}\le \dfrac{\hat{p}-p}{\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}}\le z_{\alpha/2}\right)=1-\alpha \\ P\left(\hat{p}-z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\le p \le \hat{p}+z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\right)=1-\alpha  \end{align}

The figure shows the general confidence interval on the normal curve.

How to find the multiplier using the Standard Normal Distribution

$$z_a$$ is the z-value having a tail area of $$a$$ to its right. With some calculation, one can use the Standard Normal Cumulative Probability Table to find the value.

Commonly Used Alpha Levels Section

The table is a list of frequently used alphas andtheir  $$z_{\alpha/2}$$ multipliers.

Confidence level and corresponding multiplier
Confidence Level $$\boldsymbol{\alpha}$$ $$\boldsymbol{z_{\alpha/2}}$$ $$\boldsymbol{z_{\alpha/2}}$$ Multiplier
90% .10 $$z_{0.05}$$ 1.645
95% .05 $$z_{0.025}$$ 1.960
98% .02 $$z_{0.01}$$ 2.326
99% .01 $$z_{0.005}$$ 2.576

The value of the multiplier increases as the confidence level increases. This leads to wider intervals for higher confidence levels. We are more confident of catching the population value when we use a wider interval.