Chapter 11.3, Problem 19E

a.

To determine

To represent: The data using a double box and whisker plot and to describe the shape of each distribution.

Explanation of Solution

Given Information: Relevant data regarding song downloads.

  

The median is placed at the center of the box for the data of freshmen, it is neither skewed-left nor skewed-right. Distribution is symmetric and bell-shaped.

The median is placed at the right side of the box for the data of sophomores. Most of the data lies on the right side, this means that the distribution is skewed-left.

b.

To determine

To compare: The number of songs downloaded by freshmen to that of sophomores.

Explanation of Solution

Given Information: Relevant data regarding song downloads.

Distribution of freshmen is symmetric, mean is the best measure of center. Distribution of sophomores is skewed, median is best measure for center.

Mean of freshmen $=1150$

Median of sophomores $=2000$

The median of sophomores is much more than the mean of freshmen, this signifies that generally sophomores download a lot more songs as compared to freshmen.

c.

To determine

To find: The number of freshmen who have downloaded $730-1570$ songs.

Answer to Problem 19E

Around $31$ freshmen have downloaded $730-1570$ songs on their $MP3$ players.

Explanation of Solution

Given Information: Relevant data regarding song downloads.

Calculation:

The distribution for freshmen is symmetric, it is bell-shaped. This information entails the fact that the interval $\overline{x}-\sigma$ to $\overline{x}+\sigma$ contains about $68\%$ of the data.

In this case,

  $\begin{array}{l}\overline{x}-\sigma =1150-420\\ =730\end{array}$

  $\begin{array}{l}\overline{x}+\sigma =1150+420\\ =1570\end{array}$

The interval $730-1570$ contains $68\%$ of the total freshmen,

Total number of freshmen $=45$

Freshmen who have downloaded $730-1570$ songs $=0.68\times 45$

  $\approx 31$

Thus, around $31$ freshmen have downloaded $730-1570$ songs on their $MP3$ players.

d.

To determine

To find: The number of freshmen who have downloaded $310to1990$ songs.

Answer to Problem 19E

About $95$ out of $100$ freshmen or $95\%$ of the survey size would download between $310and1990$ songs.

Explanation of Solution

Given Information: Relevant data regarding song downloads.

Calculation:

Similar to the previous part,

  $\begin{array}{l}\overline{x}-2\sigma =1150-2\times 420\\ =310\\ \overline{x}+2\sigma =1150+2\times 420\\ =1990\end{array}$

In a symmetric distribution, about $95\%$ of the data is in the $\overline{x}-2\sigma to\overline{x}+2\sigma$ interval.

Thus, near about $95$ out of $100$ freshmen or $95\%$ of the survey size would download between $310and1990$ songs.