Chapter 2, Problem 2.55SE

a.

To determine

Find the five-number summary for the given data set.

Answer to Problem 2.55SE

For generic brand.

The minimum $=24$

  $m=26$

  ${\text{Q}}_L=25$

  ${\text{Q}}_U=27.25$

The maximum $=28$

  $IQR=2.25$

For Sunmaid.

The minimum $=22$

  $m=26$

  ${\text{Q}}_L=24$

  ${\text{Q}}_U=28$

The maximum $=30$

  $IQR=4$

Explanation of Solution

Given:

The given data set is

  

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  $25,25,25,26,26,26,26,26,27,27,28,28,28$

Use five-number summary $\to$ it consists of the minimum, the lower quartile, the median, the upper quartile and the maximum.

The minimum $=25$

The median $=$ average of the two middle values (number of data values is even)$n=14$

  $n=$ total number of data values.

  $\begin{array}{l}m=\frac{7\text{th}\text{data value}+8\text{th}\text{data value}}{2}\\ =\frac{26+26}{2}\\ =\frac{52}{2}\\ =26\end{array}$

For lower quartiles.

  $0.25\left(n+1\right)=0.25\left(14+1\right)=3.75\to \text{not an integer}$

The lower quartile is the $\text{6th}$ data value increased by $\text{0}\text{.75}$ times the difference between the $\text{3rd}$ and the $\text{4th}$ data value.

  ${\text{Q}}_L\text{=25+0}\text{.75}\left(25-25\right)=25+0=25$

For upper quartiles.

  $0.75\left(n+1\right)=0.75\left(14+1\right)=11.25\to \text{not an integer}$

The lower quartile is the $\text{19th}$ data value increased by $\text{0}\text{.25}$ times the difference between the $\text{11th}$ and the $\text{12th}$ data value.

  ${\text{Q}}_U\text{=27+0}\text{.25}\left(28-27\right)=2\text{7}+0.25=27.25$

The maximum $=28$

  $IQR=Q_U-Q_L=27.25-25=2.25$

For Sunmaid.

Arrange the data set from smallest to largest.

  $22,24,24,24,24,25,25,27,28,28,28,29,30$

Use five-number summary $\to$ it consists of the minimum, the lower quartile, the median, the upper quartile and the maximum.

The minimum $=22$

The median $=$ average of the two middle values (number of data values is even)$n=14$

  $n=$ total number of data values.

  $\begin{array}{l}m=\frac{7\text{th}\text{data value}+8\text{th}\text{data value}}{2}\\ =\frac{25+27}{2}\\ =\frac{52}{2}\\ =26\end{array}$

For lower quartiles.

  $0.25\left(n+1\right)=0.25\left(14+1\right)=3.75\to \text{not an integer}$

The lower quartile is the $\text{6th}$ data value increased by $\text{0}\text{.75}$ times the difference between the $\text{3rd}$ and the $\text{4th}$ data value.

  ${\text{Q}}_L\text{=24+0}\text{.75}\left(24-24\right)=25+0=24$

For upper quartiles.

  $0.75\left(n+1\right)=0.75\left(14+1\right)=11.25\to \text{not an integer}$

The lower quartile is the $\text{19th}$ data value increased by $\text{0}\text{.25}$ times the difference between the $\text{11th}$ and the $\text{12th}$ data value.

  ${\text{Q}}_U\text{=28+0}\text{.25}\left(28-28\right)=28+0=28$

The maximum $=30$

  $IQR=Q_U-Q_L=28-24=4$

b.

To determine

Draw two box plots on the same horizontal scale.

Answer to Problem 2.55SE

The boxplot is

  

Explanation of Solution

Given:

The given data set is

  

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  $25,25,25,26,26,26,26,26,27,27,28,28,28$

Use five-number summary $\to$ it consists of the minimum, the lower quartile, the median, the upper quartile and the maximum.

The minimum $=25$

The median $=$ average of the two middle values (number of data values is even)$n=14$

  $n=$ total number of data values.

  $\begin{array}{l}m=\frac{7\text{th}\text{data value}+8\text{th}\text{data value}}{2}\\ =\frac{26+26}{2}\\ =\frac{52}{2}\\ =26\end{array}$

For lower quartiles.

  $0.25\left(n+1\right)=0.25\left(14+1\right)=3.75\to \text{not an integer}$

The lower quartile is the $\text{6th}$ data value increased by $\text{0}\text{.75}$ times the difference between the $\text{3rd}$ and the $\text{4th}$ data value.

  ${\text{Q}}_L\text{=25+0}\text{.75}\left(25-25\right)=25+0=25$

For upper quartiles.

  $0.75\left(n+1\right)=0.75\left(14+1\right)=11.25\to \text{not an integer}$

The lower quartile is the $\text{19th}$ data value increased by $\text{0}\text{.25}$ times the difference between the $\text{11th}$ and the $\text{12th}$ data value.

  ${\text{Q}}_U\text{=27+0}\text{.25}\left(28-27\right)=2\text{7}+0.25=27.25$

The maximum $=28$

  $IQR=Q_U-Q_L=27.25-25=2.25$

Outliers $\to$ it is an observation that are more than $1.5$ times the IQR above $Q_U$ or below $Q_L$ .

  $\begin{array}{l}{Q}_U+1.5IQR=27.25+1.5\left(2.25\right)=30.625\\ Q_L-1.5IQR=25-1.5\left(2.25\right)=21.625\end{array}$

There are no outliers, because all data values are between $21.625$ and $30.625$ .

For Sunmaid.

Arrange the data set from smallest to largest.

  $22,24,24,24,24,25,25,27,28,28,28,29,30$

Use five-number summary $\to$ it consists of the minimum, the lower quartile, the median, the upper quartile and the maximum.

The minimum $=22$

The median $=$ average of the two middle values (number of data values is even)$n=14$

  $n=$ total number of data values.

  $\begin{array}{l}m=\frac{7\text{th}\text{data value}+8\text{th}\text{data value}}{2}\\ =\frac{25+27}{2}\\ =\frac{52}{2}\\ =26\end{array}$

For lower quartiles.

  $0.25\left(n+1\right)=0.25\left(14+1\right)=3.75\to \text{not an integer}$

The lower quartile is the $\text{6th}$ data value increased by $\text{0}\text{.75}$ times the difference between the $\text{3rd}$ and the $\text{4th}$ data value.

  ${\text{Q}}_L\text{=24+0}\text{.75}\left(24-24\right)=25+0=24$

For upper quartiles.

  $0.75\left(n+1\right)=0.75\left(14+1\right)=11.25\to \text{not an integer}$

The lower quartile is the $\text{19th}$ data value increased by $\text{0}\text{.25}$ times the difference between the $\text{11th}$ and the $\text{12th}$ data value.

  ${\text{Q}}_U\text{=28+0}\text{.25}\left(28-28\right)=28+0=28$

The maximum $=30$

  $IQR=Q_U-Q_L=28-24=4$

Outliers $\to$ it is an observation that are more than $1.5$ times the IQR above $Q_U$ or below $Q_L$ .

  $\begin{array}{l}{Q}_U+1.5IQR=28+1.5\left(4\right)=34\\ Q_L-1.5IQR=24-1.5\left(4\right)=18\end{array}$

There are no outliers, because all data values are between $18$ and $34$ .

The boxplot starts at the lower quartile, ends at the upper quartile and vertical line at the median.

The whiskers of the boxplot are at the minimum and maximum value (excluding the outliers).

The box plot is

  

c.

To determine

Draw a stem and leaf plots for the given data sets.

Answer to Problem 2.55SE

The shape of the distributions are symmetric and boxplots confirm the result.

Explanation of Solution

Given:

The given data set is

  

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  $25,25,25,26,26,26,26,26,27,27,28,28,28$

Placethe digits of the ones to the left of the vertical line and the digits of the tenths of every data value to the right of the vertical line.

Stem	Leaf
$\begin{array}{l}24\\ 25\\ 26\\ 27\\ 28\end{array}$	$\begin{array}{l}0\\ 000\\ 0000\\ 00\\ 000\end{array}$

For Sunmaid.

Arrange the data set from smallest to largest.

  $22,24,24,24,24,25,25,27,28,28,28,29,30$

Placethe digits of the ones to the left of the vertical line and the digits of the tenths of every data value to the right of the vertical line.

Stem	Leaf
$\begin{array}{l}22\\ 23\\ 24\\ 25\\ 26\\ 27\\ 28\\ 29\\ 30\end{array}$	$\begin{array}{l}0\\ \\ 0000\\ 00\\ \\ 0\\ 0000\\ 0\\ 0\end{array}$

Both distributionsare symmetric, because most of the data values lie in the middle of the distribution.

The boxplots also confirms that the distribution is symmetric, because the box of the boxplot lies in the middle between the whiskers.

d.

To determine

Check the result for the average number of raisins for the two brands.

Answer to Problem 2.55SE

The two brands appear to have the same average number of raisins.

Explanation of Solution

Given:

The given data set is

  

Calculation:

For generic brand.

Arrange the data set from smallest to largest.

  $25,25,25,26,26,26,26,26,27,27,28,28,28$

Use five-number summary $\to$ it consists of the minimum, the lower quartile, the median, the upper quartile and the maximum.

The minimum $=25$

The median $=$ average of the two middle values (number of data values is even)$n=14$

  $n=$ total number of data values.

  $\begin{array}{l}m=\frac{7\text{th}\text{data value}+8\text{th}\text{data value}}{2}\\ =\frac{26+26}{2}\\ =\frac{52}{2}\\ =26\end{array}$

For Sunmaid.

Arrange the data set from smallest to largest.

  $22,24,24,24,24,25,25,27,28,28,28,29,30$

Use five-number summary $\to$ it consists of the minimum, the lower quartile, the median, the upper quartile and the maximum.

The minimum $=22$

The median $=$ average of the two middle values (number of data values is even)$n=14$

  $n=$ total number of data values.

  $\begin{array}{l}m=\frac{7\text{th}\text{data value}+8\text{th}\text{data value}}{2}\\ =\frac{25+27}{2}\\ =\frac{52}{2}\\ =26\end{array}$

The median of both the brands are identical, the two brands appear to have the same average number of raisins.