Chapter 14.1, Problem 13E

Teacher Data Select a cluster sample of 10 states, and find the mean of the average population and the mean of the pupils per teacher. How do your results compare with the U.S. figures?

To determine

The mean of the average population and mean of pupils per teacher.

To compare: The results with the U.S. figures(population mean).

Answer to Problem 13E

The sample mean of the average population is 51,257.1, and the sample mean of pupils per teacher is 16.69.

There is large difference between the sample and the population means in the average population, and the sample there is no large difference between the sample and the population mean in the pupils per teacher

Explanation of Solution

Calculation:

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

The data with random numbers is,

State number	States	Pupils per teacher	Average population
1	AL	15.9	47,949
2	AK	13.3	65,468
3	AZ	17.5	49,885
4	AR	15.0	46,631
5	CA	24.9	69,324
6	CO	17.6	49,844
7	CT	13.2	69,397
8	DE	14.3	59,679
9	FL	15.8	48,598
10	GA	15.7	52,880
11	HI	15.8	54,300
12	ID	18.3	49,734
13	IL	15.9	59,113
14	IN	18.6	50,065
15	IA	14.3	50,946
16	KS	13.9	47,464
17	KY	15.8	50,203
18	LA	13.8	51,381
19	ME	12.4	48,430
20	MD	14.7	64,248
21	MA	13.9	72,334
22	MI	18.4	61,560
23	MN	15.9	56,268
24	MS	15.2	41,814
25	MO	13.2	47,517
26	MT	13.4	48,855
27	NE	9.8	48,997
28	NV	18.1	55,957
29	NH	12.1	55,599
30	NJ	12.0	68,797
31	NM	15.0	45,453
32	NY	12.0	75,279
33	NC	15.1	45,737
34	ND	12.2	47,344
35	OH	17.4	56,307
36	OK	16.1	44,373
37	OR	21.8	57,612
38	PA	14.6	62,994
39	RI	13.4	63,474
40	SC	14.3	48,375
41	SD	13.8	39,018
42	TN	15.0	47,563
43	TX	15.4	48,819
44	UT	21.6	49,393
45	VT	9.2	52,526
46	VA	12.3	48,670
47	WA	19.7	52,234
48	WV	14.3	45,453
49	WI	15.5	53,797
50	WY	12.4	56,775

There are many ways to divide the population into clusters. Here, the population is divided into cluster based on ranges of pupils per teacher.

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

• In the given data the highest value of pupils per teacher is 24.9 and least value of pupils per teacher is 3.14.

$\begin{array}{c}· \text{Range }=\text{ highest value}-\text{ lowest value} \\ = 24.9-9.2\\ =15.7\end{array}$

• Divide the given data into 5 clusters, based on the range of pupils per teacher
• The range obtained is 15.7, divide the range by 5, and length of the interval would be 3.14.
• Arrangement of intervals with range 3.14 as follows.

9.2-12.34

12.35-15.48

15.49-18.62

18.63-21.76

21.77-24.9

• Arrange all the values of pupils per teacher 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	Pupils per teacher	Average population
NE	9.8	48,997
NH	12.1	55,599
NJ	12.0	68,797
NY	12.0	75,279
ND	12.2	47,344
VT	9.2	52,526
VA	12.3	48,670

Cluster2:

States	Pupils per teacher	Average population
AK	13.3	65,468
AR	15.0	46,631
CT	13.2	69,397
DE	14.3	59,679
IA	14.3	50,946
KS	13.9	47,464
LA	13.8	51,381
ME	12.4	48,430
MD	14.7	64,248
MA	13.9	72,334
MS	15.2	41,814
MO	13.2	47,517
MT	13.4	48,855
NM	15.0	45,453
NC	15.1	45,737
PA	14.6	62,994
RI	13.4	63,474
SC	14.3	48,375
SD	13.8	39,018
TN	15.0	47,563
TX	15.4	48,819
WV	14.3	45,453
WY	15.5	56,775

Cluster3:

States	Pupils per teacher	Average population
AL	15.9	47,949
AZ	17.5	49,885
CO	17.6	49,844
FL	15.8	48,598
GA	15.7	52,880
HI	15.8	54,300
ID	18.3	49,734
IL	15.9	59,113
IN	18.6	50,065
KY	15.8	50,203
MI	18.4	61,560
MN	15.9	56,268
NV	18.1	55,957
OH	17.4	56,307
OK	16.1	44,373
WI	15.5	53,797

Cluster 4:

State	Pupils per teacher	Average population
WA	19.7	52,234

Cluster 5:

State	Pupils per teacher	Average population
CA	24.9	69,324
OR	21.8	57,612
UT	21.6	49,393

From the above clusters, select cluster 3 and from that select the first 10 samples for calculating the means.

Calculate means for pupils per teacher and average population as follows.

States	Pupils per teacher	Average population
AL	15.9	47,949
AZ	17.5	49,885
CO	17.6	49,844
FL	15.8	48,598
GA	15.7	52,880
HI	15.8	54,300
ID	18.3	49,734
IL	15.9	59,113
IN	18.6	50,065
KY	15.8	50,203
Total	166.9	512,571

The sample mean of pupil per teacher is,

$\begin{array}{c}\text{Sample}\text{mean}\text{of}\text{pupils}\text{per}\text{teacher=}\frac{\text{Sum}\text{of}\text{the}\text{samples}\text{of}\text{pupils}\text{per}\text{teacher}}{\text{Number}\text{of}\text{samples}}\\ =\frac{166.9}{10}\\ =16.69\end{array}$

The population mean of pupils per teacher is

$\begin{array}{c}\text{Population}\text{mean}\text{of pupils per teacher=}\frac{\text{sum of all observations}}{\text{number of observations}}\\ =\frac{15.9+13.3+17.5+15.0+24.9+\mathrm{...}+12.4}{50}\\ =\frac{762.9}{50}\\ =15.2\end{array}$

The sample mean of average population is,

$\begin{array}{c}Samplemeanofaveragepopulation=\frac{Sumofthesamplesofaveragepopulation}{Numberofsamples}\\ =\frac{512,571}{10}\\ =51,257.1\end{array}$

The population mean for average population is

$\begin{array}{c}\text{Population}\text{mean}\text{of average population=}\frac{\text{Sum of all observations}}{\text{Number of observations}}\\ =\frac{\left(\begin{array}{l}65,468+45,631+69,397+\\ 59,946+\mathrm{...}+55,957+56,307\end{array}\right)}{50}\\ =\frac{2,684,433}{50}\\ =53,688.66\end{array}$

Thus, the sample mean of average population is 51,257.1, and the sample mean of pupils per teacher is 16.69. Also, the population mean of average population is 53,688.66, and the population mean of pupils per teacher is 15.2

Comparison of means:

It is clear that the sample there is large difference between the sample and the population means in the average population, and the sample there is no large difference between the sample and the population mean in the pupils per teacher.