Question

Questions T2.9 and T2.10 refer to the following setting. 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 $1990s$was $470$and the standard deviation was $110$. In $2009$, the mean was 515 and the standard deviation was 116 .
T2.9. What is the standardized score ( $z$-score) for a student who scored $500$ on the old SAT scale?
(a) $-30$
(b) $-0.27$
(c) $-0.13$
(d) $0.13$
(e) $0.27$

Short answer

The standardized score is $0.27$. So, the option (e) is correct.

Step-by-step solution

Given information

SAT scores were based on a scale established many years ago before the scale was altered in $1995$. In $1990$, the mean math score on the previous scale was $470$, with a standard deviation of $110$. The mean in $2009$ was $515$, with a standard deviation of $116$.

Explanation

From the old$SATs$:
$\mu =Mean=470$

$\sigma =S\tan adarddeviation=110$

$x=500$

The standardized score is the score multiplied by the standard deviation and divided by the mean:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{500-470}{110} \\ =\frac{30}{110} \\ =\frac{3}{11} \\ \approx 0.27 \end{gathered}$$