Q 8.113. - Elementary Statistics

Question

Explain why there is more variation in the possible values of the studentized version of $\overline{x}$ than in the possible values of the standardized version of $\overrightarrow{x}$

Step-by-Step Solution

Answer

Variation in $\overline{x}$ is the single cause of variation in $Z$ values, but variation in $t$ values is produced by both $\overline{x}$ and $s$ The variety in $t$ values is obviously bigger than the variation in $Z$ values.

Step 1: Given information

The standardized version of $\overline{x}$ is given by, $Z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$

Step 2: Concept

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

Step 3: Calculation

Standardized version of $\overline{x}$ is given by,

$Z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$, Where $\sigma$ is known. And studentized version of $\overline{x}$ is given by,

$t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}}$, Where $s$ is the sample S.D $s=\sqrt{\frac{1}{n-1}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}$

Step 4: Explanation

We can see from the preceding two formulas that the standardized version of $\overline{x}, Z$, is dependent on only one random variable, $\overline{x}$ The studentized form of $\overline{x}$, on the other hand, is dependent on two random variables, $\overline{x}$ and $s$ Thus, variation in the values of $Z$ is caused solely by variation in $\overline{x}$, but variation in the values of $t$ is caused by both $\overline{x}$ and $s$. Obviously, the variation in $t$ is greater than the variation in $Z$ values.