Elementary Statistics: A Step By Step Approach
Elementary Statistics: A Step By Step Approach
9th Edition
ISBN: 9780073534985
Author: Allan Bluman
Publisher: McGraw-Hill Science/Engineering/Math
Question
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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.

Expert Solution & Answer
Check Mark

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 numberStatesPupils per teacherTeacher’s average pay
1AL16.047,571
2AK14.959,672
3AZ18.946,952
4AR12.714,700
5CA21.468,203
6CO17.049,202
7CT13.264,350
8DE14.457,080
9FL15.946,708
10GA14.453,112
11HI15.255,063
12ID18.246,283
13IL14.962,077
14IN16.749,986
15IA13.849,626
16KS13.746,657
17KY15.849,543
18LA14.048,903
19ME11.146,106
20MD14.363,971
21MA13.669,273
22MI17.157,958
23MN15.452,431
24MS31.945,644
25MO13.345,317
26MT13.445,759
27NE13.446,227
28NV18.551,524
29NH12.751,443
30NJ12.065,130
31NM14.946,258
32NY11.871,633
33NC14.846,850
34ND12.142,964
35OH17.155,958
36OK15.547,691
37OR18.755,224
38PA14.059,156
39RI13.059,686
40SC14.847,508
41SD13.438,837
42TN14.746,290
43TX14.548,261
44UT22.445,885
45VT9.849,084
46VA11.750,015
47WA19.353,003
48WV14.345,959
49WI14.851,264
50WY12.355,861

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.

  Range = highest value lowest value   =  31.99.8=22.1

  • 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:

StatesPupils per teacherTeacher’s average pay
VT9.849,084
ME11.146,106
VA11.750,015
NY11.871,633
NJ12.065,130
ND12.142,964
WY12.355,861
AR12.714,700
NH12.751,443
RI13.059,686
CT13.264,350
MO13.345,317
MT13.445,759
NE13.446,227
SD13.438,837
MA13.669,273
KS13.746,657
IA13.849,626
LA14.048,903
PA14.059,156

Cluster2:

StatesPupils per teacherTeacher’s average pay
MD14.363,971
WV14.345,959
DE14.457,080
GA14.453,112
TX14.548,261
TN14.746,290
NC14.846,850
SC14.847,508
WI14.851,264
AK14.959,672
IL14.962,077
NM14.946,258
HI15.255,063
U.S.15.355,202
MN15.452,431
OK15.547,691
KY15.849,543
FL15.946,708
AL16.047,571
IN16.749,986
CO17.049,202
Ml17.157,958
OH17.155,958
ID18.246,283
NV18.551,524

Cluster3:

StatesPupils per teacherTeacher’s average pay
OR18.755,224
AZ18.946,952
WA19.353,003
CA21.468,203
UT22.445,885

Cluster 5:

StatePupils per teacherTeacher’s average pay
MS31.945,644

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.

StatesPupils per teacherAverage population
VT9.849,084
ME11.146,106
VA11.750,015
NY11.871,633
NJ12.065,130
ND12.142,964
WY12.355,861
AR12.714,700
NH12.751,443
RI13.059,686
Total119.2506,622

The sample mean of pupil per teacher is,

Samplemeanofpupilsperteacher=SumofthesamplesofpupilsperteacherNumberofsamples=119.210=11.92

The population mean of pupils per teacher is

Populationmeanof pupils per teacher=Sum of all observationsNumber of observations=16.0+14.9+18.9+...+14.3+14.8+12.350=761.750=15.234

The sample mean of teacher’s average pay is,

Samplemeanofteacher's avg pay=SumofobservationsNumberofobservations=506,62210=50,662.2

The population mean for teacher’s average pay is,

Populationmeanof teacher's avg pay=Sum of all observationsNumber of observations=(47,571+59,672+...+51,264+55,861)50=2,573,85850=51,477.16

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.

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Chapter 14 Solutions

Elementary Statistics: A Step By Step Approach

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