Chapter 14.1, Problem 19E

To determine

The sample mean of number of electoral votes.

To compare: The sample mean with the population mean.

Answer to Problem 19E

The mean of number of electoral votes is 5.9.

There is a large difference between the sample mean and population mean and this is because of the large variation between the observations.

Explanation of Solution

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 some interval values of number of electoral votes.

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

• In the data the highest value of number of electoral votes is 55 and least value of number of electoral votes is 3.

  $\begin{array}{c}\text{<x-custom-btb-me data-me-id="2244" class="microExplainerHighlight">Range</x-custom-btb-me> = highest value}-\text{lowest value}\\ =55-3\\ =52\end{array}$

• Divide the data into 5 clusters, based on the range of unemployment rates.
• The range obtained is 52.  Divide the range by 5 to get 10.4.
• Arrangement ofintervals with range 10.4 as follows.

3-13.4

13.5-23.8

23.9-34.2

34.3-44.6

44.7-55

• Arrange all the values of number of electoral votes 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:

States	Number of electoral votes
AL	9
AK	3
AZ	11
AR	6
CO	9
CT	7
DE	3
DC	3
HI	4
TD	4
TN	11
KS	6
KY	8
IA	6
LA	8
ME	4
MD	10
MA	11
MN	10
MS	6
MO	10
MT	3
NE	5
NV	6
NH	4
NM	5
ND	3
OK	7
OR	7
RI	4
SC	9
SD	3
TN	11
UT	6
VT	3
VA	13
WA	12
WV	5
WI	10
WY	3

Cluster2:

States	Number of electoral votes
GA	16
IL	20
MI	16
NJ	14
NC	15
OH	18
PA	20

Cluster3:

States	Number of electoral votes
FL	29
NY	29

Cluster4:

States	Number of electoral votes
TX	38

Cluster 5:

States	Number of electoral votes
CA	55

• From the above clusters select the first 10 observations of Cluster 1 as a sample.

States	Number of electoral votes
AL	9
AK	3
AZ	11
AR	6
CO	9
CT	7
DE	3
DC	3
HI	4
TD	4

Calculation:

To calculate mean for number of electoral votes for the selected sample is

States	Number of electoral votes
AL	9
AK	3
AZ	11
AR	6
CO	9
CT	7
DE	3
DC	3
HI	4
TD	4
Total	59

The sample mean of the number of electoral votes is,

$\begin{array}{c}samplemeanofn\text{umber of electoral votes}=\frac{Sumofthevaluesofnumberofelectorolvotes}{Numberofobservatio\sin sample}\\ =\frac{59}{10}\\ =5.9\end{array}$

The population mean is,

$\begin{array}{l}populationmean=\frac{Sumofallobservations}{Totalnumberofobservations}\\ =\frac{538}{51}\\ =10.55\end{array}$

The Sample mean of number of electoral votes is 5.9, and the population mean of number of electoral votes is10.55.

Comparison of means:

It is clear that the sample there is large difference between the sample and the population means in the average population.