Question

Consider a sample with data values of 10, 20, 12, 17, and 16. Compute the -score for each of the five observations (to 2 decimals). Enter negative values as negative numbers. Observed value -score

Answer 1

Answer:

Hence, the score for each of the five observations are $-1.25,1.25,-0.75,0.50,0.25$

Given :

Sample with data values of $x_i$  $10,20,12,17$ and $16$

Sample size$n=5$

To find:

Compute the score for each of the five observations.

Explanation :

$\because$ Sample mean $\bar{x}=\frac{\sum x_i}{n}$

                     $\Rightarrow \bar{x}=\frac{10+20+12+17+16}{5}=\frac{75}{5}$

                    $\Rightarrow \bar{x}=15$

Standard deviation $\sigma=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}$

                             $\Rightarrow \sigma=\sqrt{\frac{(10-15)^2+(20-15)^2+(12-15)^2+(17-15)^2+(16-15)^2}{5-1}}$

                            $\Rightarrow \sigma=\sqrt{\frac{(-5)^2+(5)^2+(-3)^2+(2)^2+(1)^2}{4}}$

                           $\Rightarrow \sigma=\sqrt{\frac{25+25+9+4+1}{4}}$

                          $\Rightarrow \sigma=\sqrt{\frac{64}{4}} =\sqrt{16}$

                         $\Rightarrow \sigma=4$

$\because$ The score of the observations $Z$ is $\frac{x-\bar{x}}{\sigma}$.

So, when $(x=10),$       $Z=\frac{10-15}{4}=-1.25$

     when $(x=20),$       $Z=\frac{20-15}{4}=1.25$

     when $(x=12),$      $Z=\frac{12-15}{4}=-0.75$

    when  $(x=17},$       $Z=\frac{17-15}{4}=0.50$

    when $(x=16)$       $Z=\frac{16-15}{4}=0.25$