Question

find the z - score.
{7, 9, 9, 12, 5, 3, 15, 8}
x = 2
-5.3
2.34
-1.86
5.3

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{7 + 9+9+12+5+3+15+8}{8}=\frac{68}{8}=8.5$

Step2: Calculate the standard - deviation

First, find the differences from the mean:
$(7 - 8.5)^2=(-1.5)^2 = 2.25$, $(9 - 8.5)^2=(0.5)^2 = 0.25$, $(9 - 8.5)^2=(0.5)^2 = 0.25$, $(12 - 8.5)^2=(3.5)^2 = 12.25$, $(5 - 8.5)^2=(-3.5)^2 = 12.25$, $(3 - 8.5)^2=(-5.5)^2 = 30.25$, $(15 - 8.5)^2=(6.5)^2 = 42.25$, $(8 - 8.5)^2=(-0.5)^2 = 0.25$
The sum of squared differences $S=\sum_{i = 1}^{n}(x_i-\bar{x})^2=2.25+0.25+0.25+12.25+12.25+30.25+42.25+0.25 = 100$
The standard - deviation $s=\sqrt{\frac{S}{n - 1}}=\sqrt{\frac{100}{7}}\approx3.78$

Step3: Calculate the z - score

The formula for the z - score is $z=\frac{x-\bar{x}}{s}$. Substitute $x = 2$, $\bar{x}=8.5$ and $s\approx3.78$ into the formula.
$z=\frac{2 - 8.5}{3.78}=\frac{-6.5}{3.78}\approx - 1.72$ (There may be some rounding - off differences in the given options). The closest value to our calculated result among the options is - 1.86.

Answer:

-1.86