Question

which standard deviation would you prefer in a class distribution with μ = 50, when your score is x = 54 on a chemistry exam?
○ σ = 2
○ σ = 4
○ this is impossible to determine based on the information provided.
○ σ = 8

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{X - \mu}{\sigma}$, where $X$ is the individual score, $\mu$ is the mean, and $\sigma$ is the standard deviation. A higher z - score indicates a better relative position in the distribution.

Step2: Calculate z - scores for each option

For $\sigma = 2$: $z_1=\frac{54 - 50}{2}=\frac{4}{2}=2$.
For $\sigma = 4$: $z_2=\frac{54 - 50}{4}=\frac{4}{4}=1$.
For $\sigma = 8$: $z_3=\frac{54 - 50}{8}=\frac{4}{8}=0.5$.

Step3: Compare z - scores

Since $z_1>z_2>z_3$, a standard deviation of $\sigma = 2$ gives the highest z - score, meaning the score of 54 is in a better relative position compared to other options.

Answer:

$\sigma = 2$