Elementary Statistics: A Step By Step Approach
Elementary Statistics: A Step By Step Approach
10th Edition
ISBN: 9781259755330
Author: Allan G. Bluman
Publisher: McGraw-Hill Education
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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?

Expert Solution & Answer
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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.

·        Range = highest value lowest value   =  24.99.2=15.7

  • 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,

Samplemeanofpupilsperteacher=SumofthesamplesofpupilsperteacherNumberofsamples=166.910=16.69

The population mean of pupils per teacher is

Populationmeanof pupils per teacher=sum of all observationsnumber of observations=15.9+13.3+17.5+15.0+24.9+...+12.450=762.950=15.2

The sample mean of average population is,

Samplemeanofaveragepopulation=SumofthesamplesofaveragepopulationNumberofsamples=512,57110=51,257.1

The population mean for average population is

Populationmeanof average population=Sum of all observationsNumber of observations=(65,468+45,631+69,397+59,946+...+55,957+56,307)50=2,684,43350=53,688.66

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.

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Chapter 14 Solutions

Elementary Statistics: A Step By Step Approach

Ch. 14.1 - List the advantages and disadvantages of cluster...Ch. 14.1 - Teacher Data Using the table of random numbers,...Ch. 14.1 - Prob. 12ECh. 14.1 - Teacher Data Select a cluster sample of 10 states,...Ch. 14.1 - Record High Temperatures Which method of sampling...Ch. 14.1 - Prob. 16ECh. 14.1 - Electoral Votes Select a systematic sample of 10...Ch. 14.1 - Electoral Votes Divide the 50 states into five...Ch. 14.1 - Prob. 19ECh. 14.1 - Define sampling or selection bias.Ch. 14.1 - Prob. 22ECh. 14.1 - Define nonresponsive bias.Ch. 14.1 - Prob. 24ECh. 14.1 - Define response or interview bias.Ch. 14.1 - Prob. 26ECh. 14.1 - Define volunteer bias.Ch. 14.1 - Give an example of how volunteer bias might occur.Ch. 14.2 - Prob. 1ACCh. 14.2 - Prob. 1ECh. 14.2 - Prob. 2ECh. 14.2 - Prob. 3ECh. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Prob. 6ECh. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Exercises 1 through 9 include questions that...Ch. 14.2 - Prob. 11ECh. 14.2 - Exercises 1 through 9 include questions that...Ch. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Exercises 1 through 9 include questions that...Ch. 14.2 - Prob. 17ECh. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.3 - Simulations Answer the following questions: 1....Ch. 14.3 - Prob. 1ECh. 14.3 - Prob. 2ECh. 14.3 - Who is responsible for the development of modern...Ch. 14.3 - Prob. 4ECh. 14.3 - Prob. 5ECh. 14.3 - Prob. 6ECh. 14.3 - Prob. 7ECh. 14.3 - Prob. 8ECh. 14 - Hurricanes Select a random sample of eight storms...Ch. 14 - Prob. 14.1.2RECh. 14 - Hurricanes Select a cluster of 10 storms. Compute...Ch. 14 - Prob. 14.1.4RECh. 14 - Prob. 14.1.5RECh. 14 - Prob. 14.1.6RECh. 14 - Prob. 14.1.8RECh. 14 - Prob. 14.2.9RECh. 14 - Prob. 14.2.10RECh. 14 - Prob. 14.2.11RECh. 14 - Prob. 14.2.12RECh. 14 - Prob. 1DACh. 14 - Prob. 2DACh. 14 - Select a cluster sample of 20 individuals, and...Ch. 14 - Prob. 4DACh. 14 - Prob. 5DACh. 14 - Determine whether each statement is true or false....Ch. 14 - Prob. 2CQCh. 14 - Prob. 3CQCh. 14 - Prob. 4CQCh. 14 - Prob. 5CQCh. 14 - Prob. 6CQCh. 14 - Interviewing selected people at a local...Ch. 14 - Prob. 8CQCh. 14 - Prob. 9CQCh. 14 - Prob. 10CQCh. 14 - Blood Pressure Select a random sample of 12...Ch. 14 - Prob. 12CQCh. 14 - Blood Pressure Divide the individuals into...Ch. 14 - Prob. 14CQCh. 14 - Prob. 25CQCh. 14 - Prob. 26CQCh. 14 - Prob. 27CQCh. 14 - Prob. 28CQCh. 14 - For Exercises 2530, explain why the survey...Ch. 14 - Prob. 30CQCh. 14 - Prob. 1DP
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