Presidents. From the Information Please Almanac, we compiled the following table on U.S. region of birth and political party of the first 44 U.S. presidents. The table uses these abbreviations:

D=Democratic, Republican, D=Democratic, W=Whig, F=Federalist, DR=Democratic-Republican, D=Democratic, Republican, D=Democratic, NE=Northeast, MW=Midwest, SO=South, WE=West; R=Republicans, U=Union;

Region	Party	Region	Party	Region	Party
SO	F	SO	R	MW	R
NE	F	SO	U	NE	D
SO	DR	MW	R	MW	D
SO	DR	MW	R	SO	R
SO	DR	MW	R	NE	D
NE	DR	NE	R	SO	D
SO	D	NE	D	WE	R
NE	D	MW	R	MW	R
SO	W	NE	D	SO	D

SO	W	MW	R	MW	R
SO	D	NE	R	NE	R
SO	W	MW	R	SO	D
NE	W	SO	D	NE	R
NE	D	MW	R	WE	D
NE	D	NE	R

a. What is the population under consideration?

b. What are the two variables under consideration?

c. Group the bivariate data for the variables "birth region" and "party" into a contingency table.

A quantitative data set has been grouped by using cut point grouping with equal-width classes of with 8 .

a)

If the class's midpoint is 10, we can calculate the lower and higher cut points as follows:

The midpoint of a class is defined as

$$\begin{gathered} Midpoint=\frac{Lowercutpoint+Uppercutpoint}{2} \\ 10=\frac{Lowercutpoint+Uppercutpoint}{2} \\ Lowercutpoint+uppercutpoint=20....\left(1\right) \end{gathered}$$

Given that the width of the class interval is 8

That is,

$Uppercutpoint-Lowercutpoint=8$.....(2)

When (1) and (2) are combined, we get

$$\begin{gathered} 2Uppercutpoint=28 \\ Uppercutpoint=\frac{28}{2} \\ =14 \end{gathered}$$

By substituting the upper cut point in equation (1), we get

$$\begin{gathered} Lowercutpoint+Uppercutpoint=20 \\ Lowercutpoint=20-Uppercutpoint \\ =20-14 \\ =6 \end{gathered}$$

Therefore, The first class is 6.14

b) We find the class midpoint of the second class is shown below:

The Lower cut point of the first class is 6 .

The first class's upper cut point is 14.

In Cut point grouping the lower cut point of the second class is equal to the upper cut point of the first class.

So the Lower cut point of the second class is 14

The upper cut point of the second class is

= Lower cut point of the second class + Class width

=14+8

=22

By definition, the midpoint is given by

$$\begin{gathered} \mathrm{Midpoint}=\frac{\mathrm{Lower}\mathrm{cutpoint}+\mathrm{Upper}\mathrm{cutpoint}}{2} \\ =\frac{14+22}{2} \\ =\frac{36}{2} \\ =18 \end{gathered}$$

Therefore, the midpoint of the second class is 18 .

c) We find the lower and cutpoints of the third class is shown below:

The Lower cutpoint of the second class is 14 .

The Upper cutpoint of the second class is 22 .

In Cutpoint grouping the lower cutpoint of the third class is equal to the upper cutpoint of the second class.

So the Lower cutpoint of the third class is 22

The upper cutpoint of the third class is

= Lower cutpoint of the third class + Class width

$$\begin{gathered} =22+8 \\ =30 \end{gathered}$$

Therefore, the lower and upper cutpoints of the third class is 22 and 30 .

d)The third class's lower and higher cutpoints are 22 and 30 respectively.

Therefore the third class constants an observation 22.