Chapter 11.5, Problem 20PPS

(a)

To determine

To find: The $z$ values that corresponds to her score on each test.

Answer to Problem 20PPS

The z-values are $1.03$ , $0.806$ , and $1.207$ .

Explanation of Solution

Given:

The given table is as follows.

	Math	Science	Social science
$\mu$	$76$	$81$	$72$
$\sigma$	$9.7$	$6.2$	$11.6$

Calculation:

Calculate the corresponding $z$ value for Math.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{86-76}{9.7}\\ =1.03\end{array}$

Calculate the corresponding $z$ value for Science.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{86-81}{6.2}\\ =0.806\end{array}$

Calculate the corresponding $z$ value for Social science.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{86-72}{11.6}\\ =1.207\end{array}$

Therefore, the z-values are $1.03$ , $0.806$ , and $1.207$ .

(b)

To determine

To find: The probability of a student scoring an 86 or lower on each test.

Answer to Problem 20PPS

The probabilities are $0.84850$ , $0.78988$ , and $0.88628$ .

Explanation of Solution

Given:

The given table is as follows.

	Math	Science	Social science
$\mu$	$76$	$81$	$72$
$\sigma$	$9.7$	$6.2$	$11.6$

Calculation:

Find the probability of a student scoring an 86 or lower on Math.

  $\begin{array}{c}P\left(X\le 86\right)=P\left(z\le 1.03\right)\\ =0.84850\end{array}$

Find the probability of a student scoring an 86 or lower on Science.

  $\begin{array}{c}P\left(X\le 86\right)=P\left(z\le 0.806\right)\\ =0.78988\end{array}$

Find the probability of a student scoring an 86 or lower on Social Science.

  $\begin{array}{c}P\left(X\le 86\right)=P\left(z\le 1.207\right)\\ =0.88628\end{array}$

Therefore, the probabilities are $0.84850$ , $0.78988$ , and $0.88628$ .

(c)

To determine

To find: The test in which Nikki’s score is highest, and explain reasoning.

Answer to Problem 20PPS

On social Studies test, Nikki’s standardized score was highest.

Explanation of Solution

Given:

The given table is as follows.

	Math	Science	Social science
$\mu$	$76$	$81$	$72$
$\sigma$	$9.7$	$6.2$	$11.6$

Calculation:

On social Studies test, Nikki’s standardized score was highest.

The mean of Social Studies test is minimum in all three tests, so Nikki’s standardized score is highest in social Studies.

Therefore, on social Studies test, Nikki’s standardized score was highest.