Chapter 6, Problem 42E

a)

To determine

To find the IQ score for the 15^th percentile.

Answer to Problem 42E

The IQ score for the 15^th percentile is 83.36

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

Formula:

  $x=\mu +z\sigma$

  P(Z < z) = 0.15

  z = -1.04

Using formula,

  $x=100+(-1.04)\times 16=83.36$

Hence, the IQ score for the 15^th percentile is 83.36

b)

To determine

To find the IQ score for the 98^th percentile.

Answer to Problem 42E

The IQ score for the 98^th percentile is 132.8

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

Formula:

  $x=\mu +z\sigma$

  P(Z < z) = 0.98

  z = 2.05

Using formula,

  $x=100+2.05\times 16=132.8$

Hence, the IQ score for the 98^th percentile is 132.8

c)

To determine

To find the IQR.

Answer to Problem 42E

The IQR is 21.44

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

Formula:

  $x=\mu +z\sigma$

First, we need to find first quartile and third quartile. Therefore, we need to find z-score corresponding to a probability of 0.25 or 0.75.

  z = $\pm 0.67$

Using formula,

  $\begin{array}{l}{Q}_1=100+(-0.67)\times 16=89.28\\ Q_3=100+0.67\times 16=110.72\end{array}$

So, $IQR=110.72-89.28=21.44$