Chapter 6, Problem 40E

a)

To determine

To find the cutoff values for the highest 5% of all IQs.

Answer to Problem 40E

The cutoff values for the highest 5% of all IQs are 126.32

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

Formula:

  $x=\mu +z\sigma$

  P(Z > z) = 0.05

  P(Z< z) = 1-0.05 = 0.95

  z = 1.645

Using formula,

  $x=100+1.645\times 16=126.32$

Hence, cutoff values for the highest 5% of all IQs are 126.32

b)

To determine

To find the cutoff values for the lowest 30% of all IQs.

Answer to Problem 40E

The cutoff values for the lowest 30% of all IQs are 91.68

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

Formula:

  $x=\mu +z\sigma$

  P(Z < z) = 0.30

  z = -0.52

Using formula,

  $x=100+(-0.52)\times 16=91.68$

Hence, cutoff values for the lowest 30% of all IQs are 91.68

c)

To determine

To find the cutoff values for the middle 80% of all IQs.

Answer to Problem 40E

The cutoff values for the middle 80% of all IQs are 91.68

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

Formula:

  $x=\mu +z\sigma$

  P(-z <Z < z) = 0.80

  z = $\pm 1.28$

Using formula,

  $\begin{array}{l}x=100+(-1.28)\times 16=79.52\\ x=100+1.28\times 16=120.48\end{array}$

Hence, cutoff values for the middle 80% of all IQs are 79.52 and 120.48