Respuesta :
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]
