Question

Until the scale was changed in $1995$SAT scores were based on a scale set many years ago. For Math scores, the mean under the old scale in the early $1990s$was $470$and the standard deviation was $110$ In $2016$ the mean was $510$and the standard deviation was $103$ Gina took the SAT in $1994$and scored $500$Her cousin Colleen took the SAT in $2016$ and scored $530$ Who did better on the exam, and how can you tell?

a. Colleen—she scored $30$points higher than Gina.

b. Colleen—her standardized score is higher than Gina’s.

c. Gina—her standardized score is higher than Colleen’s.

d. Gina—the standard deviation was larger in $2016$

e. The two cousins did equally well—their z-scores are the same.

Short answer

The correct option is (c) Gina—her standardized score is higher than Colleen’s.

Step-by-step solution

Given information

${\mu }_G=470$

${\sigma }_G=110$

$x_G=500$

${\mu }_C=510$

${\sigma }_C=103$

$x_C=530$

Concept

$z=\frac{x-\mu }{\sigma }$

Calculation

$$\begin{gathered} z_G=\frac{x-\mu }{\sigma } \\ =\frac{500-470}{110} \\ =0.27 \\ {z}_C=\frac{x-\mu }{\sigma } \\ =\frac{530-510}{103} \\ =0.19 \end{gathered}$$

It has been discovered that $z_G>z_C$ implying that the person with the greatest $z-$score will perform higher on SAT exam in comparison to the year in which they took the exam.

Hence, the correct option is (c)