Chapter 7, Problem R7.6RE

(a)

To determine

To Calculate: the probability that a randomly chosen individual has a WAIS score of 105 or greater.

Answer to Problem R7.6RE

0.3707

Explanation of Solution

Given:

  $\begin{array}{c}\mu =100\\ \sigma =15\end{array}$

Formula used:

  $z=\frac{x-\mu }{\sigma }$

Calculation:

The z-score is

  $\begin{array}{c}=\frac{x-\mu }{\sigma }\\ =\frac{105-100}{15}\\ =0.33\end{array}$

The associatively probability using table

  $\begin{array}{c}P\left(X\ge 105\right)=P\left(z>0.33\right)\\ =P\left(z<-0.33\right)\\ =0.3707\end{array}$

(b)

To determine

To find: the mean and standard deviation of the sampling distribution of the average WAIS score $\overline{x}$ for an SRS of 60 people.

Answer to Problem R7.6RE

Mean = 100

Standard deviation = 1.9365

Explanation of Solution

Given:

  $\begin{array}{c}\mu =100\\ \sigma =15\\ n=60\end{array}$

Formula used:

  ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$

Calculation:

The mean of the sampling distribution of the sample mean $\overline{x}$ is

  ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}=\frac{15}{\sqrt{60}}=1.9365$

(c)

To determine

To Calculate: the probability that the average WAIS score of an SRS of 60 people is 105 or greater.

Answer to Problem R7.6RE

0.0049

Explanation of Solution

Given:

  $\begin{array}{c}\mu =100\\ \sigma =15\\ n=60\end{array}$

Formula used:

  $\begin{array}{c}{\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ z=\frac{x-\mu }{\sigma }\end{array}$

Calculation:

The mean of the sampling distribution of the sample mean $\overline{x}$ is same to the population standard deviation divided by the square root of the sample size

  ${\mu }_{\overline{x}}=\mu =100$

The standard deviation of the sampling distribution of the sample mean $\overline{x}$ is

  ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}=\frac{15}{\sqrt{60}}=1.9365$

The z-score is

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{105-100}{1.9365}\\ =2.58\end{array}$

The corresponding probability using table

  $\begin{array}{c}P\left(\overline{x}\ge 105\right)=P\left(z>2.58\right)\\ =P\left(z<-2.58\right)\\ =0.0049\end{array}$

(d)

To determine

To find: the affect if the distribution of WAIS scores in the adult population was distinctly non-normal.

Answer to Problem R7.6RE

1. Affected
2. and (c) not affected

Explanation of Solution

The answer in (a) could be very different, because it is supposed that the population distribution was normal to find the probability.

The answer in (b) and (c) will not be very different, the reason is that the sample size is 30 or more and, by the central limit theorem, know that the sampling distribution is about normal even if the population distribution is not normal.