Chapter 6, Problem 26E

a)

To determine

To draw the model which shows 68-95-99.7

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

Following is the model which shows 68-95-99.7:

  

b)

To determine

To find the interval in which central 95% of IQ scores to be found.

Answer to Problem 26E

The interval in which central 95% of IQ scores to be found between 68 and 132

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

According to rule of 68-95-99.7, the 95% of the values are within 2 standard deviation of the mean.

Therefore,

  $\begin{array}{l}\mu -2\sigma =100-2\times 16=132\\ \mu +2\sigma =100+2\times 16=68\end{array}$

Hence, interval in which central 95% of IQ scores to be found between 68 and 132

c)

To determine

To find the percent of people having IQ score above 116.

Answer to Problem 26E

The 16% percent of people having IQ score above 116.

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

According to rule of 68-95-99.7,

  $\mu +\sigma =100+16=116$

That means, 116 is one standard deviation above the mean. We know that 68% of the IQ scores are within 1 standard deviation of the mean. That is, 100-68 = 32% of the IQ scores are more than 1 standard deviation from the mean.

We know, the normal curve is symmetric about the mean then 16% is more than 1 standard deviation below the mean and 16% is more than 1 standard deviation above the mean.

Hence, approximately 16% of all people should have an IQ score above 116.

d)

To determine

To find the percent of people having IQ score between 68 and 84

Answer to Problem 26E

The 13.5% percent of people having IQ score between 68 and 84.

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

According to rule of 68-95-99.7,

  $\mu +\sigma =100+16=116$

That means, 116 is one standard deviation above the mean.

  $\mu -2\sigma =100-2\times 16=68$

That means, 68 is two standard deviation below the mean.

Therefore, 95%-68% = 27% which is more than 1 standard deviation and less than 2 standard deviation of the mean. We know, the normal curve is symmetric about the mean then 13.5% is more than 1 standard deviation but less than 2 standard deviation below the mean and 13.5% is more than 1 standard deviation but less than 2 standard deviation above the mean. So, we can say, 13.5% of people having IQ score between 68 and 84.

e)

To determine

To find the percent of people having IQ score above 132

Answer to Problem 26E

The 13.5% percent of people having IQ score above 132

Explanation of Solution

Given:

  $\mu$ = 100

  $\sigma$ = 16

According to rule, 95% of the data lies between 2 standard deviations from the mean. Since, total data is 100%, so 5% of the data is the more than 2 standard deviations from the mean. Therefore, as per symmetry, 2.5% is more than 2 standard deviations below the mean and 2.5% is more than 2 standard deviations above the mean.

  $\mu +2\sigma =100+2\times 16=132$

That means, 132 is two standard deviation above the mean.

Therefore, approximately 2.5% of people having IQ score above 132