Chapter 14.1, Problem 13E

To determine

The mean of the average teacher’s pay and mean of pupils per teacher.

To compare: The results with the U.S. figures.

Answer to Problem 13E

The sample mean of the average population is 50,662.2 and the sample mean of pupils per teacher is 11.92

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 serial numbers is,

State number	States	Pupils per teacher	Teacher’s average pay
1	AL	16.0	47,571
2	AK	14.9	59,672
3	AZ	18.9	46,952
4	AR	12.7	14,700
5	CA	21.4	68,203
6	CO	17.0	49,202
7	CT	13.2	64,350
8	DE	14.4	57,080
9	FL	15.9	46,708
10	GA	14.4	53,112
11	HI	15.2	55,063
12	ID	18.2	46,283
13	IL	14.9	62,077
14	IN	16.7	49,986
15	IA	13.8	49,626
16	KS	13.7	46,657
17	KY	15.8	49,543
18	LA	14.0	48,903
19	ME	11.1	46,106
20	MD	14.3	63,971
21	MA	13.6	69,273
22	MI	17.1	57,958
23	MN	15.4	52,431
24	MS	31.9	45,644
25	MO	13.3	45,317
26	MT	13.4	45,759
27	NE	13.4	46,227
28	NV	18.5	51,524
29	NH	12.7	51,443
30	NJ	12.0	65,130
31	NM	14.9	46,258
32	NY	11.8	71,633
33	NC	14.8	46,850
34	ND	12.1	42,964
35	OH	17.1	55,958
36	OK	15.5	47,691
37	OR	18.7	55,224
38	PA	14.0	59,156
39	RI	13.0	59,686
40	SC	14.8	47,508
41	SD	13.4	38,837
42	TN	14.7	46,290
43	TX	14.5	48,261
44	UT	22.4	45,885
45	VT	9.8	49,084
46	VA	11.7	50,015
47	WA	19.3	53,003
48	WV	14.3	45,959
49	WI	14.8	51,264
50	WY	12.3	55,861

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 31.9 and least value of pupils per teacher is 9.8.

$\begin{array}{c} \text{Range }=\text{ highest value}-\text{ lowest value} \\ = 31.9-9.8\\ =22.1\end{array}$

• Divide the given data into 5 clusters, based on the range of pupils per teacher
• The range obtained is 22.1, divide the range by 5, and length of the interval would be 4.42.
• Arrangement of intervals with range 4.42 as follows.

9.8-14.22

14.23-18.65

18.66-23.08

23.09-27.51

27.52-31.94

• 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	Teacher’s average pay
VT	9.8	49,084
ME	11.1	46,106
VA	11.7	50,015
NY	11.8	71,633
NJ	12.0	65,130
ND	12.1	42,964
WY	12.3	55,861
AR	12.7	14,700
NH	12.7	51,443
RI	13.0	59,686
CT	13.2	64,350
MO	13.3	45,317
MT	13.4	45,759
NE	13.4	46,227
SD	13.4	38,837
MA	13.6	69,273
KS	13.7	46,657
IA	13.8	49,626
LA	14.0	48,903
PA	14.0	59,156

Cluster2:

States	Pupils per teacher	Teacher’s average pay
MD	14.3	63,971
WV	14.3	45,959
DE	14.4	57,080
GA	14.4	53,112
TX	14.5	48,261
TN	14.7	46,290
NC	14.8	46,850
SC	14.8	47,508
WI	14.8	51,264
AK	14.9	59,672
IL	14.9	62,077
NM	14.9	46,258
HI	15.2	55,063
U.S.	15.3	55,202
MN	15.4	52,431
OK	15.5	47,691
KY	15.8	49,543
FL	15.9	46,708
AL	16.0	47,571
IN	16.7	49,986
CO	17.0	49,202
Ml	17.1	57,958
OH	17.1	55,958
ID	18.2	46,283
NV	18.5	51,524

Cluster3:

States	Pupils per teacher	Teacher’s average pay
OR	18.7	55,224
AZ	18.9	46,952
WA	19.3	53,003
CA	21.4	68,203
UT	22.4	45,885

Cluster 5:

State	Pupils per teacher	Teacher’s average pay
MS	31.9	45,644

From the above clusters, select cluster 1 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
VT	9.8	49,084
ME	11.1	46,106
VA	11.7	50,015
NY	11.8	71,633
NJ	12.0	65,130
ND	12.1	42,964
WY	12.3	55,861
AR	12.7	14,700
NH	12.7	51,443
RI	13.0	59,686
Total	119.2	506,622

The sample mean of pupil per teacher is,

$\begin{array}{c}\text{Sample}\text{mean}\text{of}\text{pupils}\text{per}\text{teacher}=\frac{\text{Sum}\text{of}\text{the}\text{samples}\text{of}\text{pupils}\text{per}\text{teacher}}{\text{Number}\text{of}\text{samples}}\\ =\frac{119.2}{10}\\ =11.92\end{array}$

The population mean of pupils per teacher is

$\begin{array}{c}\text{Population}\text{mean}\text{of pupils per teacher}=\frac{\text{Sum of all observations}}{\text{Number of observations}}\\ =\frac{16.0+14.9+18.9+\mathrm{...}+14.3+14.8+12.3}{50}\\ =\frac{761.7}{50}\\ =15.234\end{array}$

The sample mean of teacher’s average pay is,

$\begin{array}{c}\text{Sample}\text{mean}\text{of}\text{teacher's avg pay}=\frac{\text{Sum}\text{of}\text{observations}}{\text{Number}\text{of}\text{observations}}\\ =\frac{506,622}{10}\\ =50,662.2\end{array}$

The population mean for teacher’s average pay is,

$\begin{array}{c}\text{Population}\text{mean}\text{of teacher's avg pay}=\frac{\text{Sum of all observations}}{\text{Number of observations}}\\ =\frac{\left(47,571+59,672+\mathrm{...}+51,264+55,861\right)}{50}\\ =\frac{2,573,858}{50}\\ =51,477.16\end{array}$

Thus, the sample mean of teacher’s average pay is 50,662.2 and the sample mean of pupils per teacher is 11.92. Also the population mean of teacher’s average pay is 51,477.16, and the population mean of pupils per teacher is 15.234.

Comparison of means:

In U.S., the number of pupils per teacher is 15.3 and the teacher’s average pay is 55,202.

There is large difference between the number of pupils per teacher in the U.S. and the sample mean and also in the average of teacher’s pay of the sample is less than the U.S. teacher’s average pay.

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.