Chapter 11.5, Problem 1GP

(a)

To determine

To find: The range of values that represent the middle $95\%$ of the distribution.

Answer to Problem 1GP

The range is $5.6<X<10.8$ .

Explanation of Solution

Given:

The normal distribution has a mean of $8.2$ and a standard deviation of $1.3$ .

Calculation:

The middle $95\%$ of data in a normal distribution is the range from $\mu -2\sigma \text{to}\mu +2\sigma$ .

Therefore, the range of values in the middle $95\%$ is calculated as:

  $\begin{array}{c}8.2-2\left(1.3\right)<X<8.2+2\left(1.3\right)\\ 5.6<X<10.8\end{array}$

The range is $5.6<X<10.8$ .

(b)

To determine

To find: The percent of the data will be less than $4.3$ .

Answer to Problem 1GP

  $0.15\%$ of the data will be less than $4.3$ .

Explanation of Solution

As $4.3$ is $3\sigma$ more than $\mu$ .

  $99.7\%$ of the data fall between $\mu -3\sigma \text{and}\mu +3\sigma$ , so the remaining data values represented by the two tails covers $0.3\%$ of the distribution. We are only concerned with the lower tail, so $0.15\%$ of the data will be less than $4.3$ .

Therefore $0.15\%$ of the data will be less than $4.3$ .