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:

Hence, the score for each of the five observations are [tex]-1.25,1.25,-0.75,0.50,0.25[/tex]

Given :

Sample with data values of [tex]x_i[/tex]  [tex]10,20,12,17[/tex] and [tex]16[/tex]

Sample size[tex]n=5[/tex]

To find:

Compute the score for each of the five observations.

Explanation :

[tex]\because[/tex] Sample mean [tex]\bar{x}=\frac{\sum x_i}{n}[/tex]

                     [tex]\Rightarrow \bar{x}=\frac{10+20+12+17+16}{5}=\frac{75}{5}[/tex]

                    [tex]\Rightarrow \bar{x}=15[/tex]

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

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

                            [tex]\Rightarrow \sigma=\sqrt{\frac{(-5)^2+(5)^2+(-3)^2+(2)^2+(1)^2}{4}}[/tex]

                           [tex]\Rightarrow \sigma=\sqrt{\frac{25+25+9+4+1}{4}}[/tex]

                          [tex]\Rightarrow \sigma=\sqrt{\frac{64}{4}} =\sqrt{16}[/tex]

                         [tex]\Rightarrow \sigma=4[/tex]

[tex]\because[/tex] The score of the observations [tex]Z[/tex] is [tex]\frac{x-\bar{x}}{\sigma}[/tex].

So, when [tex](x=10),[/tex]       [tex]Z=\frac{10-15}{4}=-1.25[/tex]

     when [tex](x=20),[/tex]       [tex]Z=\frac{20-15}{4}=1.25[/tex]

     when [tex](x=12),[/tex]      [tex]Z=\frac{12-15}{4}=-0.75[/tex]

    when  [tex](x=17},[/tex]       [tex]Z=\frac{17-15}{4}=0.50[/tex]

    when [tex](x=16)[/tex]       [tex]Z=\frac{16-15}{4}=0.25[/tex]

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