Chapter 14, Problem 14.1.8RE

To determine

The mean of unemployment rates by using cluster sampling and the mean of average weekly benefits by using cluster sampling.

To compare: The sample mean with the U.S. figures.

Answer to Problem 14.1.8RE

Mean of unemployment rates is 8.29, and mean of the average weekly benefits is 301.8.

By comparing the result with the U.S. figures, it is clear that there is slight difference between the cluster sample mean and the U.S. figures of unemployment rates and the cluster sample mean and the U.S. figures are very close or approximately equal in values in case of weekly average benefits.

Explanation of Solution

Given info:

The unemployment rate and average weekly benefit for all the states in U.S. are provided.

Calculation:

Answers will vary. One of the possible answers is given below:

There are many ways to divide the population into clusters to ensure maximum heterogeneity within each cluster. Here, the population is divided into 5 clusters of 10 states each, alphabetically. Heterogeneity is preserved in existing alphabetical arrangement of data.

Procedure for selecting the 10 samples by using cluster sampling is as follows.

• Take the 1^st 10 states as Cluster1,
• Take the 2^nd set of 10 states as Cluster2,
• Take the 3^rd set of 10 states as Cluster,
• Take the 4^th set of 10 states as Cluster4 and
• Take the 5^th set of 10 states as Cluster5

Cluster1:

States

AL

AK

AZ

AR

CA

CO

CT

DE

FL

GA

Cluster2:

States

HI

ID

IL

IN

IA

KS

KY

LA

ME

MD

Cluster3:

States

MA

MI

MN

MS

MO

MT

NE

NV

NH

NJ

Cluster4:

States

NM

NY

NC

ND

OH

OK

OR

PA

RI

SC

Cluster5:

States

SD

TN

TX

UT

VT

VA

WA

WV

WI

WY

• From the figure 14.1(Table of random numbers), select a starting point for randomly selecting a cluster.
• Here, the selected starting point is ‘2’ which is present in sixth row and first column.
• Thus, the randomly chosen cluster is Cluster2.

To calculate means for unemployment rates and mean of average weekly benefits for the selected Cluster2.

Cluster2:

State	Unemployment rates	Weekly average benefits
HI	6.6	416
ID	9.3	255
IL	10.3	317
IN	10.2	295
IA	6.1	321
KS	7.0	326
KY	10.5	289
LA	7.5	209
ME	7.9	274
MD	7.5	316

The sample mean of unemployment rate is,

$\begin{array}{c}Samplemeanofunemploymentrates=\frac{Sumofthesamplesofunemploymentrates}{Numberofsamplesinunemploymentrate}\\ =\frac{6.6+9.3+\mathrm{...}+7.9+7.5}{10}\\ =\frac{82.9}{10}\\ =\mathrm{8.29.}\end{array}$

The U.S. figure for unemployment rate is 9.6.

The sample mean of average weekly benefit is,

$\begin{array}{c}Samplemeanofaverageweeklybenfit=\frac{Sumofthesamplesaverageweeklybenefits}{Numberofsamplesinweeklyaveragebenefits}\\ =\frac{416+255+\mathrm{...}+274+316}{10}\\ =\frac{3018}{10}\\ =\mathrm{301.8.}\end{array}$

The U.S. figure for average weekly benefits is 299.

Comparisons of means:

The sample mean of unemployment rates is 8.29, and the U.S. figure for unemployment rate is 9.6. That implies that the sample mean is slightly lesser than the population mean. It is clear that there is slight difference between the sample mean and the U.S. figures.

Again, the sample mean of the average weekly benefits is 301.8, and the U.S. figure for average weekly benefits is 299. It is clear that the sample mean and the U.S. figures are very close or approximately equal in values.