Question

The test scores for the analytical writingsection of a particular standardized test canbe approximated by a normal distribution, asshown in the figure.(a) What is the maximum score that can be inthe bottom 10% of scores?(b) Between what two values does the middle80% of scores lie?

Answer 1

Solution:

Given:

Recall that the z-value is expressed as

$\begin{gathered} z=\frac{x-\mu}{\sigma} \\ \text{where} \\ \mu\Rightarrow\operatorname{mean}\text{ value} \\ \sigma\Rightarrow s\tan dard\text{ deviation} \end{gathered}$

Thus,

$z=\frac{x-3.7}{0.91}\text{ ---- equation 1}$

A) maximum score that can be in the bottom 10% of scores

using the table of z-values,

for the bottom 10% scores, we have

$z=-1.28155156554$

To evaluate x, substitute the value of z into equation 1.

Thus,

$\begin{gathered} -1.28155156554=\frac{x-3.7}{0.91}\text{ } \\ \Rightarrow x=2.5337895 \end{gathered}$

Thus, the maximum score that can be in the bottom 10% of scores is 2.5

B) Two values for which the middle 80% of scores lie.

From the z score values shown below:

The z scores of the value are

$\begin{gathered} z_1=-1.28 \\ z_2=1.28 \end{gathered}$

Thus,

$\begin{gathered} \text{when z=-1.28, we have} \\ -1.28=\frac{x-3.7}{0.91}\text{ } \\ \Rightarrow x=2.5352 \\ \text{when z=1.28, we have} \\ 1.28=\frac{x-3.7}{0.91} \\ \Rightarrow x=4.8648 \end{gathered}$

Thus, the two values for which the middle 80% of scores lie are 2.5 and 4.86.