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 4.95, and mean of the average weekly benefits is 336.1

By comparing the result with the U.S. figures, it is clear that the there is large difference between the sample of the unemployment rate and the population unemployment rate. And also that the there is large difference between the sample of the weekly benefits and the population of weekly 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. Here the population is divided into cluster based on ranges of unemployment rates.

• Procedure for selecting the 10 samples by using cluster sampling is as follows.
• In the data the highest unemployment rate is 9.8 and least unemployment rate is 2.9.

$\begin{array}{c}\text{<x-custom-btb-me data-me-id="2244" class="microExplainerHighlight">Range</x-custom-btb-me>}=\text{highest value}-\text{lowest value}\\ =9.8-2.9\\ =6.9\end{array}$

• Divide the data into 5 clusters, based on the range of unemployment rates.
• The range obtained is 6.9.  Divide the range by 5, to get 1.38.
• Arrangement ofintervals with range 1.38 as follows.

2.9-4.28

4.29-5.66

5.67-7.04

7.05-8.42

8.42-9.8

• Arrange all the values of unemployment rates according to the intervals and each interval is named as cluster1, cluster2, cluster3, cluster4, cluster5.
• The following table represents the 5 clusters, which are arranged according to the intervals.

Cluster1:

State	Unemployment rates	Weekly average benefits
NE	3.9	276
ND	2.9	396
SD	3.8	276

Cluster2:

State	Unemployment rates	Weekly average benefits
HI	4.8	424
IA	4.6	337
KS	5.4	341
MN	5.1	367
NH	5.3	287
OK	5.4	293
UT	4.4	345
VT	4.4	313
VA	5.5	295
WY	4.6	359

Cluster3:

States	Unemployment rates	Weekly average benefits
AL	6.5	207
AK	6.5	250
CO	6.8	256
DE	6.7	245
ID	6.2	246
LA	6.2	207
ME	6.7	285
MD	6.6	329
MO	6.5	242
MT	5.6	290
NM	6.9	303
TX	6.3	241
WA	7.0	387
WV	6.5	275
WI	6.7	276

Cluster4:

States	Unemployment rates	Weekly average benefits
AZ	8.0	221
AR	7.5	289
CT	7.8	348
FL	7.2	231
GA	8.2	267
IN	7.5	243
MA	7.1	424
KY	8.3	292
NJ	8.2	398
NY	7.7	308
NC	8.0	290
OH	7.4	318
OR	7.7	316
PA	7.4	360
SC	7.6	248
TN	8.2	235

Cluster 5:

states	Unemployment rates	Average weekly benefits
CA	8.9	301
IL	9.2	324
MI	8.8	293
MS	8.6	194
NV	9.8	308
RI	9.5	351

• From the above clusters select the 2^nd cluster as a sample.

To calculate means for unemployment rates and mean of average weekly benefits for the selected 2^nd cluster.

Cluster2:

State	Unemployment rates	Weekly average benefits
HI	4.8	424
IA	4.6	337
KS	5.4	341
MN	5.1	367
NH	5.3	287
OK	5.4	293
UT	4.4	345
VT	4.4	313
VA	5.5	295
WY	4.6	359

The sample mean of unemployment is,

$\begin{array}{c}samplemeanofunemploymentrates=\frac{Sumofthesamplesofunemploymentrates}{Numberofsamplesinunemploymentrate}\\ =\frac{49.5}{10}\\ =4.95\end{array}$

The population mean of unemployment is,

$\begin{array}{c}\text{Mean}\text{of}\text{unemployment}\text{rates}=\frac{6.5+6.5+8.0+7.5+8.9+6.8+7.8+\mathrm{........}+6.7+4.6}{50}\\ =\frac{336.4}{50}\\ =6.728\end{array}$

The sample mean of average weekly benfit is,

$\begin{array}{c}Samplemeanofaverageweeklybenfit=\frac{Sumofthesamplesaverageweeklybenefits}{Numberofsamplesinweeklyaveragebenefits}\\ =\frac{3,361}{10}\\ =336.1\end{array}$

The population mean of average weekly benefits is,

$\begin{array}{c}\text{Mean}\text{of}\text{average}\text{weekly}\text{benefits}=\frac{\left(\begin{array}{l}207+250+221+289+301+356+\\ 345+\mathrm{...}+275+276+359\end{array}\right)}{50}\\ =\frac{15,122}{50}\\ =302.44\end{array}$

Comparisons of means:

The sample mean of unemployment rates is 4.95, and the population mean of unemployment rates is 6.728. That implies that the sample mean is lesser than the population mean. It is clear that there is larger difference between the sample mean and the population mean.

Also, the sample mean of the average weekly benefits is 336.1, and the population mean of pupils per teacher is 302.44. That implies that the sample mean is greater than the population mean. It is clear that there is larger difference between the sample mean and the population mean.