Chapter 11.1, Problem 33E

(a)

To determine

To calculate:

The percentage of SAT math score

Answer to Problem 33E

The percentage of SAT math score is $88.49\%$ .

Explanation of Solution

Given information:

  $\begin{array}{l}\mu =514\\ \sigma =118\\ x=650\end{array}$

Formula used:

  $Z-$ Score is given by the following expression:

  $Z=\frac{x-\mu }{\sigma }$

Where,

Standard deviation: $\sigma$

Mean: $\mu$

Calculation:

The percentage of SAT math score is calculated by using the $Z-$ Score formula:

  $\begin{array}{l}P\left(x=650\right)=P\left(z=\frac{650-514}{118}\right)\\ P\left(x=650\right)=P\left(z=1.1525\right)\end{array}$

Use the standard normal table:

  $P\left(z=1.1525\right)=P\left(z\approx 1.12\right)=0.8849=88.49\%$

(b)

To determine

To calculate:

The percentage of ACT math score

Answer to Problem 33E

The percentage of ACT math score is $93.32\%$ .

Explanation of Solution

Given information:

  $\begin{array}{l}\mu =21.0\\ \sigma =5.3\\ x=29\end{array}$

Formula used:

  $Z-$ Score is given by the following expression:

  $Z=\frac{x-\mu }{\sigma }$

Where,

Standard deviation: $\sigma$

Mean: $\mu$

Calculation:

The percentage of ACT math score is calculated by using the $Z-$ Score formula:

  $\begin{array}{l}P\left(x=29\right)=P\left(z=\frac{29-21}{5.3}\right)\\ P\left(x=29\right)=P\left(z=1.5094\right)\end{array}$

Use the standard normal table:

  $P\left(z=1.5094\right)=0.9332=93.32\%$

(c)

To determine

To explain:

The better performance of the student is in ACT or SAT

Answer to Problem 33E

The student performed better in CAT math

Explanation of Solution

The students scores $650$ and $29$ in SAT and CAT math respectively. That means the student scores $88.49\%$ and $93.32\%$ in the respective subjects.

On comparing both subjects:

  $93.32\%$ is higher than $88.49\%$ . Therefore, the student performs better in CAT math.