Question

A variable has a mean of $100$ and a standard deviation of $16$ Four observations of this variable have a mean of $108$ and a sample standard deviation of $12$ . Determine the observed value of the

a. standardized version of $\overline{x}$.

b. studentized version of $\overline{x}$.

Short answer

Part (a) $1$

Part (b) $1.33$

Step-by-step solution

Part (a) Step 1: Given information

Sample size $n=4$

Sample mean $\overline{x}=108$

Sample s.d., $s=12$

Part (a) Step 2: Concept

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

Part (a) Step 3: Calculation

Let the variable $x$has meant, $\mu =100$ and s.d., $\sigma =16$

A standardized version of $\overline{x}$,

$$\begin{gathered} Z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}} \\ =\frac{108-100}{\frac{16}{\sqrt{4}}}=\frac{8}{16/2}=\frac{8}{8}=1 \end{gathered}$$

Part (b) Step 1: Calculation

Studentized version of $\overline{x}$

$$\begin{gathered} t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}} \\ =\frac{108-100}{\frac{12}{\sqrt{4}}} \\ =\frac{8}{12/2}=\frac{8}{6}=1.33 \end{gathered}$$