Chapter 5, Problem 4P

For a population with $\mu =60$and $\sigma =12$

1. Find the a-score for each of the following X values. (Note: You should be able to find these values using the definition of a z-score. You should not need to use a formula or do any serious calculations.)

   $\begin{array}{l}X=75X=48X=84\\ X=54X=78X=51\end{array}$

Find the score (X value) that corresponds to each of the following z-scores. (Again, you should not need a formula or any serious calculations.)

   $\begin{array}{l}z=1.00z=0.25z=1.50\\ z=-0.50z=-1.25z=-2.50\end{array}$

To determine

1. For given mean and standard deviation (SD) find Z score for given X values.
2. To find corresponding X value for given Z scores.

Answer to Problem 4P

Solution:

a) For given $mean=60$ and $standarddeviation=12$ and given X values Z scores were calculated and the answers are presented in table format.

Table (a)

mean	60
sd	12
X(given)	Z=(X-mean)/sd
75	1.25
48	-1
84	2
54	-0.5
78	1.5
51	-0.75

b) For given $mean=60$ and $standarddeviation=12$ and given Z scores, X values were calculated and the answers are presented in table 4b.

Table (b)

mean	60
sd	12
Z(Given)	X(=Z* sd+mean)
1	72
0.25	63
1.5	78
-0.5	54
-1.25	45
-2.5	30

Explanation of Solution

Given:

a) Populations with $mean=60$ and $standarddeviation=12$ and some given X values.

b) Populations with $mean=60$ and $standarddeviation=12$ and some given Z scores.

Key Point:

a) In order to calculate Z scores using the given mean, SD (standard deviation) and X values we have to use the formula for Z scores which is

$Z=\left(X-mean\right)/sd$

b) In order to calculate X scores using the given mean, SD and Z values we have to use the formula:

$X=Z\times sd+mean$

Calculation:

a) $mean=60,sd=12$

Let's calculate Z for $X=75$ (other values are calculated in similar fashion and

Presented in Table (a)

$\begin{array}{l} Z=\left(X-mean\right)/sd=\left(75-60\right)/12=1.25\hfill \\ \text{For }X=48,Z=\left(48-60\right)/12=-1\hfill \\ X=84,Z=\left(84-60\right)/12=2\hfill \\ X=54,Z=\left(54-60\right)/12=-.5\hfill \\ X=78,Z=\left(78-60\right)/12=1.5\hfill \\ X=51,Z=\left(51-60\right)/12=-.75\hfill \end{array}$

b) $mean=60,sd=12$

Let's calculate X for $Z=1$ (other values are calculated in similar fashion and presented in Table (b)

$\begin{array}{l}X=Z\times sd+mean=12+60=72\hfill \\ \text{For }Z=0.25,X=.25\times 12+60=63\hfill \\ Z=1.5,X=1.5\times 12+60=78\hfill \\ Z=-.5,X=-.5\times 12+60=54\hfill \\ Z=-1.25,X=-1.25\times 12+60=45\hfill \\ Z=-2.5,X=-2.5\times 12+60=30\hfill \end{array}$

Conclusion:

Z score for given X values are calculated and given in Solution (Table a) and X values for given Z scores are calculated and presented in Solution (Table b).