Question

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 - score formula

The z - score formula is $z=\frac{X - M}{s}$, where $X$ is the raw score, $M$ is the mean, and $s$ is the standard deviation. A higher (more positive) z - score indicates a better performance relative to others.

Step2: For part a, when $X = 35$, $M = 40$

Case 1: $s = 2$

$z_1=\frac{35 - 40}{2}=\frac{- 5}{2}=-2.5$

Case 2: $s = 8$

$z_2=\frac{35 - 40}{8}=\frac{-5}{8}=-0.625$
Since $-0.625>-2.5$, the performance is better when $s = 8$.

Step3: For part b, when $X = 46$, $M = 40$

Case 1: $s = 2$

$z_3=\frac{46 - 40}{2}=\frac{6}{2}=3$

Case 2: $s = 8$

$z_4=\frac{46 - 40}{8}=\frac{6}{8}=0.75$
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 $s = 2$.