Question

1. on an exam with a mean of m = 40, you obtain a score of x = 35.

a. relative to other students, would your performance on the exam be better with a standard deviation of s = 2 or with a standard deviation of s = 8? (hint: sketch each distribution and find the location of your score.)
b. if your score were x = 46, would you prefer s = 2 or s = 8? explain your answer.

Explanation:

Step1: Calculate z - scores for part a

The z - score formula is $z=\frac{X - M}{s}$. For $X = 35$, $M = 40$, when $s = 2$, $z_1=\frac{35 - 40}{2}=\frac{- 5}{2}=-2.5$. When $s = 8$, $z_2=\frac{35 - 40}{8}=\frac{-5}{8}=-0.625$. A higher (less negative) z - score means a better relative performance. Since $-0.625>-2.5$, the performance is better when $s = 8$.

Step2: Calculate z - scores for part b

For $X = 46$, $M = 40$, when $s = 2$, $z_3=\frac{46 - 40}{2}=\frac{6}{2}=3$. When $s = 8$, $z_4=\frac{46 - 40}{8}=\frac{6}{8}=0.75$. A higher z - score means a better relative performance. Since $3>0.75$, the performance is better when $s = 2$.

Answer:

a. The performance is better with $s = 8$.
b. The performance is better with $s = 2$.