Question

In $2005,1,475,623$students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean $\mu =520$and standard deviation $\sigma =115$

1. Calculate the z-score for an SAT score of $720$. Interpret it using a complete sentence.
2. What math SAT score is 1.5 standard deviations above the mean? What can you say about this SAT score?
3. For 2012, the SAT math test had a mean of $514$ and standard deviation $117$. The ACT math test is an alternate to the SAT and is approximately normally distributed with mean $21$ and standard deviation $5.3$. If one person took the SAT math test and scored $700$ and a second person took the ACT math test and scored $30$, who did better with respect to the test they took?

Short answer

1. The z- score of $720$is $1.74$standard deviation above the mean value.
2. The value of the z- score $1.5$is $692.5$.
3. The z-score for the ACT score is greater than the z-score for the SAT score. So second person did better than the first person.

Step-by-step solution

Given information (Part a)

Given in the question that

Normal distribution with Mean $=520$

Standard diviation$=115$

Solution (Part a)

Here we need to calculate the z-score of $x=720$

The calculation is given below,

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

$=\frac{720-520}{115}$

$=1.74$

Given information (Part b)

Given in the question that

Mean$=520$

Standard deviation$=115$

Solution (Part b)

Here we need to find the SAT math's score of $1.5$standard deviations above the mean,

So the calculation is,

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

$1.5=\frac{x-520}{115}$

$x=1.5\times 115+520$

$=692.5$

Given information (Part c)

Given in the question that

Mean$=520$

Standard deviation$=115$

Solution (part c)

We need to calculate the z-score for both tests to get the desired result.

Here is the z-score for the SAT score is calculated below,

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

$=\frac{700-514}{117}$

$=1.59$

So, the z-score for the ACT score will be,

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

$=\frac{30-21}{5.3}$

$=1.70$