Question

Caitlin took both the SAT and the ACT college entrance exams. Her scores on both exams are shown in the table, as well as the national mean scores and standard deviations. Exam Caitlin's Exam Score Mean Exam Score Standard Deviation SAT 1850 1500 250 ACT 28 20.8 4.8 She wants to know on which test she performed better. Find the z-scores for her result on each exam.

Answer 1

Answer:

On the SAT her z-score was 1.4

On the ACT her z-score was 1.5

Due to the higher z-score, she performed better on the ACT.

Step-by-step explanation:

The z-score measures how many standard deviations a score X is above or below the mean. it is given by the following formula:

$Z = \frac{X - \mu}{\sigma}$

In which $\mu$ is the mean and $\sigma$ is the standard deviation.

In this problem

We find her z-score for both the SAT and the ACT.

SAT

Exam Caitlin's Exam Score Mean Exam Score Standard Deviation

SAT    1850                          1500                        250

So $X = 1850, \mu = 1500, \sigma = 250$

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1850 - 1500}{250}$

$Z = 1.4$

ACT

Exam Caitlin's Exam Score Mean Exam Score Standard Deviation

ACT    28                              20.8                         4.8

So $X = 28, \mu = 20.8, \sigma = 4.8$

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{28 - 20.8}{4.8}$

$Z = 1.5$

She wants to know on which test she performed better. Find the z-scores for her result on each exam.

On the SAT her z-score was 1.4

On the ACT her z-score was 1.5

Due to the higher z-score, she performed better on the ACT.