Chapter 2, Problem 69CE

a.

To determine

Obtain a sample of 10 pages from this textbook by using simple random sampling.

Answer to Problem 69CE

The sample of 10 pages obtained from this textbook by using simple random sampling consists of the pages serial numbered as follows:

586	559	98	206	375
73	127	263	10	843

Explanation of Solution

Calculation:

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

This textbook is considered as the population. The total number of pages in this textbook is 865 (from cover to cover).

Assign serial numbers 1 to 865 to the pages, starting with 1 for the first cover page and proceeding thereafter.

Simple random sampling:

Simple random sampling is defined as a method of sampling in which, a sample of size n is drawn from a population of size N by using a random method, such as a random number table or software such as EXCEL, MINITAB, etc. to ensure that each of the N population units has the same probability of being selected in each draw. The simple random sampling gives a sample that is representative of the population.

EXCEL Procedure:

Step by step procedure to obtain a simple random sample by using EXCEL is given below:

• Enter =RANDBETWEEN(1,865) in cells A1:A10 to obtain 10 random numbers lying between 1 and 865.
• Repeat the procedure till 10 unique random numbers are obtained.

Output using EXCEL is given below:

Hence, the sample of 10 pages obtained from this textbook by using simple random sampling consists of the pages serial numbered as follows:

586	559	98	206	375
73	127	263	10	843

b.

To determine

Obtain a sample of 10 pages from this textbook by using systematic sampling.

Answer to Problem 69CE

The sample of 10 pages obtained from this textbook by using systematic sampling consists of the pages serial numbered as follows:

82	168	254	340	426
512	598	684	770	856

Explanation of Solution

Calculation:

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

Systematic sampling:

Systematic sampling is defined as a method of sampling in which, at first, an entity is chosen from the first k entities in the list, followed by choosing every k^th entity thereafter, till the desired sample size is obtained. It is a random sampling method.

For a population size N and sample size n, $k\approx \frac{N}{n}$.

Here, $N=865$, $n=10$. Thus,

$\begin{array}{c}\frac{N}{n}=\frac{865}{10}\\ =86.5\\ k\approx 86.\end{array}$

EXCEL Procedure:

Step by step procedure to obtain a systematic sample by using EXCEL is given below:

• Enter =RANDBETWEEN(1,86) in cell A1 to obtain 1 random number lying between 1 and 86.

Output using EXCEL is given below:

Denote $r_1=82$. Add 86 successively to this random number $r_1$, till 10 distinct numbers are obtained.

Here, $r_i=r_{i-1}+86,i=2,3,\mathrm{...},10$. The calculations are shown in the following table:

$r_i=r_{i-1}+86$
82
168
254
340
426
512
598
684
770
856

Hence, the sample of 10 pages obtained from this textbook by using systematic sampling consists of the pages serial numbered as follows:

82	168	254	340	426
512	598	684	770	856

c.

To determine

Obtain a sample of 10 pages from this textbook by using cluster sampling.

Answer to Problem 69CE

The sample of 10 pages obtained from this textbook by using cluster sampling consists of the pages serial numbered as follows:

141	142	143	144	145
146	147	148	149	150

Explanation of Solution

Calculation:

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

Cluster sampling:

Cluster sampling is defined as a method of sampling in which, at first, the entire population is divided into heterogeneous subgroups, especially by geographic areas called cluster, then desired number of clusters are selected from all the clusters, usually by simple random sampling. Each cluster is expected to be representative of the population.

Divide the book into clusters of 10 consecutive pages, starting from the first page. There are 865 pages. Thus, the total number of clusters would be as follows:

• 86 clusters of 10 pages each.
• 1 cluster of 5 pages.

Assign serial numbers 1 to 87 to the 87 clusters. Randomly select 1 cluster and consider all the pages in that cluster as the sample.

EXCEL Procedure:

Step by step procedure to obtain a cluster sample by using EXCEL is given below:

Step by step procedure to obtain a systematic sample by using EXCEL is given below:

• Enter =RANDBETWEEN(1,87) in cell A1 to obtain 1 random number lying between 1 and 87.

Output using EXCEL is given below:

Thus, the 15^th cluster is selected.

Now,

The 1^st cluster has page numbers 1 to 10.

The 2^nd cluster has page numbers 11 to 20.

Proceeding in this manner,

The 86^th cluster has page numbers 851 to 860.

The 87^th cluster has page numbers 861 to 865.

The pattern above ensures that the i^th cluster must have page numbers (10i – 9) to 10i for $i=1,2,\mathrm{...},86$.

For the 15^th cluster, $i=15$. Thus,

$\begin{array}{c}10i=10\times 15\\ =150.\end{array}$

$\begin{array}{c}10i-9=150-9\\ =141.\end{array}$

Hence, the sample of 10 pages obtained from this textbook by using cluster sampling consists of the pages serial numbered as follows:

141	142	143	144	145
146	147	148	149	150

d.

To determine

Obtain a sample of 10 pages from this textbook by using judgment sampling.

Answer to Problem 69CE

The sample of 10 pages obtained from this textbook by using judgment sampling consists of the pages serial numbered as follows:

29	49	83	195	277
361	463	571	675	720

Explanation of Solution

Calculation:

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

Judgment sampling:

Judgment sampling is defined as a method of sampling that depends upon the expertise of the researcher to identify sample units that would represent the population, rather than on random sampling methods. It is a non-random sampling method.

It is considered to be a good judgment that in this particular case that the third page of each of chapters 1, 2, 3, 5, 7, 9, 11, 13, 15 and 17 would be a good representative of all the pages.

The chapters and the page number of their corresponding third pages are given below by a proper inspection of the table:

Chapter number	Third page number
1	29
2	49
3	83
5	195
7	277
9	361
11	463
13	571
15	675
17	720

Note that the page numbers here are not the number marked at the top of the page, but the actual page number of the entire pdf.

Hence, the sample of 10 pages obtained from this textbook by using judgment sampling consists of the pages serial numbered as follows:

29	49	83	195	277
361	463	571	675	720

e.

To determine

Recommend a method to estimate the mean number of formulas per page.

Answer to Problem 69CE

The method of simple random sampling may be recommended to estimate the mean number of formulas per page.

Explanation of Solution

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

Simple random sampling provides a sample that is more-or-less representative of the population. In order to estimate the mean number of formulas per page, one would prefer that the sample represents the population well enough to ensure a reasonable estimate for the whole textbook.

Thus, the method of simple random sampling may be recommended to estimate the mean number of formulas per page.

In case of systematic sampling, there may be some periodicity observed based on the nature of the contents of each chapter, depending upon the random start. The same pattern of too few formulas or too many formulas may be observed in systematic sampling. Such a periodicity is not desirable. Thus, systematic sampling is not used in this case.

Cluster sampling can be useful when the population itself forms heterogeneous clusters naturally, especially in case of geographical regions. Here, although the population forms natural heterogeneous subgroups of chapters, each of those subgroups contains more than 10 pages. Now, in case of cluster sampling, the randomly selected cluster is considered entirely into the sample. In this case, it is not possible. Thus, cluster sampling is not used in this case.

Judgment sampling is non-random sampling methods. Now, it is known that a random sampling method is more likely to yield a well-representative sample than non-random sampling methods. As a result, the use of judgment sampling is usually avoided, unless it is absolutely impossible or unreasonable to use random sampling methods. Here, it is not impossible to use random sampling. Thus, judgment sampling is not used in this case.