Chapter 2, Problem 2.55CE

The breakdown of U.S. cities having a population of at least 10,000 persons has been reported as follows. Source: Bureau of the Census, Statistical Abstract of the United States 2009, p.34.

a. How many cities have a population of a least 25,000 but less than 500,000?

b. How many cities have a population less than 2.50,000?

c. How many cities have a population of at least 100,000 but less than 1,000,000? What percentage of cities are in this group?

d. What is the class mark for the 100,000—under 250,000 class?

e. Convert the table to a relative frequency distribution.

To determine

(a)

To find:

The number of cities that have a population of at least 25,000 but less than 500,000.

Answer to Problem 2.55CE

1342.

Explanation of Solution

Given:

Frequency distribution is:

Population	Number of Cities
10,000- under 25,000	1510
25,000 -under 50,000	684
50,000 -under 100,000	430
100,000 - under 250,000	191
250,000 - under 500,000	37
500,000- under 1,000,000	25
1,000,000 or more	9

The number of cities that have a population of at least 25,000 but less than 500,000 can be calculated as:

  $\begin{array}{c}\left(\begin{array}{l}\text{The number of cities that have a }\\ \text{population of at least 25,000 }\\ \text{but less than 500,000}\end{array}\right)=684+430+191+37\\ =1342\end{array}$

Hence, the required number is 1342.

To determine

(b)

To find:

The number of cities that have population of less than 250,000.

Answer to Problem 2.55CE

2815.

Explanation of Solution

The number of cities that have a population less than 250,000 can be calculated as:

  $\begin{array}{c}\left(\begin{array}{l}\text{The number of cities that have a }\\ \text{population less than 250,000}\end{array}\right)=1510+684+430+191\\ =2815\end{array}$

Hence, the required number is 2815.

To determine

(c)

To find:

The number of cities that have a population of at least 100,000 but less than 1,000,000. Also, find the percentages of cities in this group.

Answer to Problem 2.55CE

253 cities and 8.77%.

Explanation of Solution

The number of cities that have a population of at least 100,000 but less than 1,000,000 can be calculated as:

  $\begin{array}{c}\left(\begin{array}{l}\text{The number of cities that have a }\\ \text{population of }100,000\\ \text{ but less than 1,000,000}\end{array}\right)=191+37+25\\ =253\end{array}$

The percentage of cities in this group be calculated as:

  $\begin{array}{c}\text{Required percentage = }\frac{\left(\begin{array}{l}\text{The number of cities that have a }\\ \text{population of }100,000\\ \text{ but less than 1,000,000}\end{array}\right)}{\text{Total population}}\times 100\\ =\frac{253}{1510+684+430+191+37+25+9}\times 100\\ =\frac{253}{2886}\times 100\\ =8.766\%\end{array}$

To determine

(d)

To find:

The class mark for the class 100,000-under 250,000.

Answer to Problem 2.55CE

175000.

Explanation of Solution

The class-mark or mid-point of an interval is calculated as:

  $\text{Class mark}=\frac{\text{Upper class limit}+\text{Lower class limit}}{2}$

The class mark for the class 100,000-under 250,000can be calculated as:

  $\begin{array}{c}\text{Class mark}=\frac{250000+100000}{2}\\ =\frac{350000}{2}\\ =175000\end{array}$

To determine

(d)

To find:

A relative frequency distribution for the provided data set.

Explanation of Solution

Population	Relative frequency
10,000- under 25,000	0.523
25,000 -under 50,000	0.237
50,000 -under 100,000	0.149
100,000 - under 250,000	0.066
250,000 - under 500,000	0.013
500,000- under 1,000,000	0.009
1,000,000 or more	0.003