Chapter 8.1, Problem 8.16E

(a)

To determine

To calculate the probability that a randomly chosen individual has a WAIS score of $115$ or higher.

Answer to Problem 8.16E

  $0.1587$

Explanation of Solution

Given:

  $\begin{array}{l}\mu =1000\\ \sigma =15\end{array}$

The standardized score is as:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{115-100}{15}\\ \approx 1\end{array}$

Determine the corresponding probability using the normal probability table:

  $\begin{array}{l}P(X>115)=P(Z>1)\\ =1-P(Z<1)\\ =1-0.8413\\ =0.1587\end{array}$

(b)

To determine

To calculate in what range do the score of middle $95\%$ of the adult population lie.

Answer to Problem 8.16E

Lies between $70.6\text{ and }129.4$ .

Explanation of Solution

The middle $95\%$ has the property that $2.5\%$ is below the lower boundary and $2.5\%$ is above the upper boundary. Note that the upper boundary then also has the property that $97.5\%$ is below the upper boundary.

Determine the z -score in the normal probability table that corresponds with the probability of $2.5\%$ and $97.5\%$ .

  $z=\pm 1.96$

The standardized score is as:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ x=\mu +2\sigma \\ x=100-1.96(15)=70.6\\ x=100+1.96(15)=129.4\end{array}$

We then note that the middle $95\%$ have a WAIS score between $70.6\text{ and }129.4$ .