Chapter 6, Problem 21T

Solution

a.

To State:

The z-score of the score of the given student.

  $0.5$

Given:

Scores on standardized test administrated to 10000 students have a mean of 50 and a standard deviation of 10.

Kevin has a score $55$ .

Concepts Used:

The z-score of a particular data point $x$ is $z=\frac{x-\overline{x}}{\sigma }$ . Here, $\overline{x}$ is the mean and $\sigma$ is the standard deviation.

Calculations:

Calculate the z-score for $x=55$ , $\overline{x}=50$ , and, $\sigma =10$ .

  $\begin{array}{c}z=\frac{x-\overline{x}}{\sigma }\\ =\frac{55-50}{10}\\ =0.5\end{array}$

Conclusion:

Kevin has a z-score of $0.5$ .

b.

To State:

The z-score of the score of the given student.

  $2$

Given:

Scores on standardized test administrated to 10000 students have a mean of 50 and a standard deviation of 10.

Manuel has a score $70$ .

Concepts Used:

The z-score of a particular data point $x$ is $z=\frac{x-\overline{x}}{\sigma }$ . Here, $\overline{x}$ is the mean and $\sigma$ is the standard deviation.

Calculations:

Calculate the z-score for $x=70$ , $\overline{x}=50$ , and, $\sigma =10$ .

  $\begin{array}{c}z=\frac{x-\overline{x}}{\sigma }\\ =\frac{70-50}{10}\\ =2\end{array}$

Conclusion:

Manuel has a z-score of $2$ .

c.

To State:

The z-score of the score of the given student.

  $-1$

Given:

Scores on standardized test administrated to 10000 students have a mean of 50 and a standard deviation of 10.

Colby has a score $40$ .

Concepts Used:

The z-score of a particular data point $x$ is $z=\frac{x-\overline{x}}{\sigma }$ . Here, $\overline{x}$ is the mean and $\sigma$ is the standard deviation.

Calculations:

Calculate the z-score for $x=40$ , $\overline{x}=50$ , and, $\sigma =10$ .

  $\begin{array}{c}z=\frac{x-\overline{x}}{\sigma }\\ =\frac{40-50}{10}\\ =-1\end{array}$

Conclusion:

Colby has a z-score of $-1$ .

d.

To State:

The z-score of the score of the given student.

  $-0.3$

Given:

Scores on standardized test administrated to 10000 students have a mean of 50 and a standard deviation of 10.

Neal has a score $47$ .

Concepts Used:

The z-score of a particular data point $x$ is $z=\frac{x-\overline{x}}{\sigma }$ . Here, $\overline{x}$ is the mean and $\sigma$ is the standard deviation.

Calculations:

Calculate the z-score for $x=47$ , $\overline{x}=50$ , and, $\sigma =10$ .

  $\begin{array}{c}z=\frac{x-\overline{x}}{\sigma }\\ =\frac{47-50}{10}\\ =-0.3\end{array}$

Conclusion:

Neal has a z-score of $-0.3$ .