Question

the table below shows the scores on a math test.

	david	ashley	andrew	monica	lucas
gina	75	60	75	100	75
kei
anna
jennifer

the number of standard deviations between a score and the mean score is indicated by a z - score. what was ashleys z - score?
a. 1.1
b. 60
c. - 1.1
d. - 15.5

Explanation:

Step1: Calculate the mean

First, sum all scores: $75 + 75+95 + 60+65+75+60+75+100+75=750$. There are $n = 10$ scores. The mean $\bar{x}=\frac{750}{10}=75$.

Step2: Calculate the standard - deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
$(75 - 75)^{2}=0$, $(75 - 75)^{2}=0$, $(95 - 75)^{2}=400$, $(60 - 75)^{2}=225$, $(65 - 75)^{2}=100$, $(75 - 75)^{2}=0$, $(60 - 75)^{2}=225$, $(75 - 75)^{2}=0$, $(100 - 75)^{2}=625$, $(75 - 75)^{2}=0$.
The sum $\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=400+225 + 100+225+625=1575$.
$s=\sqrt{\frac{1575}{9}}\approx\sqrt{175}\approx13.23$.

Step3: Use the z - score formula

The z - score formula is $z=\frac{x-\bar{x}}{s}$. We know that the number of standard deviations between a score $x$ and the mean $\bar{x}$ is given by the z - score. We want to find Ashley's z - score. Ashley's score $x = 60$, $\bar{x}=75$, and $s\approx13.23$.
$z=\frac{60 - 75}{13.23}=\frac{- 15}{13.23}\approx - 1.1$.

Answer:

c. - 1.1