Question

Why do you need to consider the student zed version of $\overline{x}$ to develop a confidence-interval procedure for a population means when the population standard deviation is unknown?

Short answer

$t$ does not depend on population $s.d.$, $\sigma$ and follows $t$ distribution with $df=n-1$

Step-by-step solution

Concept

The formula used: $Z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$

Explanation

The $Z-$interval procedure of finding confidence interval of $\mu$ is based on the standardized version of$\overline{x},Z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$ and hence the C.I of $\mu$ depends on $\sigma$. But, if $\sigma$ is unknown then we can not use the $Z-$interval procedure based on $Z$. In this situation we consider studentized version of $\overline{x}$, $t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}}$ and use one mean $t-$interval procedure of finding confidence interval of $\mu$ based on $t$, since studentized variable $t$ does not depend on population s.d., $\sigma$ and follows $t$ distribution with $df=n-1$