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Chapter 14: Problem 16

An audit is performed on last year's 15,000 student-aid packages given out by the financial aid office at Tasmania State University. Roughly half of the student-aid packages were less than \(\$ 1000\) (Category 1 ), about one-fourth were between \(\$ 1000\) and \(\$ 5000\) (Category 2 ), and another quarter were over \(\$ 5000\) (Category 3 ). For each audit described below, name the sampling method that best describes it. Choose your answer from the following: (A) simple random sampling, (B) convenience sampling, (C) quota sampling. (D) stratified sampling, (E) census. (a) The auditor reviews all 15,000 student-aid packages. (b) The auditor randomly selects 200 student-aid packages in Category 1,100 student-aid packages in Category 2 and 100 transactions in Category \(3 .\) (c) The auditor reviews the first 500 student-aid packages that he comes across. (d) The auditor first separates the student- aid packages by school (Agriculture, Arts and Humanities, Engineering. Nursing, Social Science, Science, and Mathematics). Three of these schools are selected at random and further subdivided by major. Ten majors are randomly seIected within each selected school, and then 20 students are randomly selected from each of the selected majors.

Short Answer

Expert verified
The correct sampling methods are (a) census (E), (b) stratified sampling (D), (c) convenience sampling (B), and (d) could be considered a form of cluster sampling which is not listed in the given options. Typically, cluster sampling would be the closest choice.

Step by step solution

01

- Understanding Census

A census is a systematic method that collects data from every member of the population. When an auditor reviews all 15,000 student-aid packages, it means each and every package is investigated without sampling.
02

- Identifying Stratified Sampling

Stratified sampling involves dividing the population into subgroups, or strata, and then taking a random sample from each subgroup. In the case where the auditor selects a certain number of packages from each category, this is a perfect example of stratified sampling.
03

- Recognizing Convenience Sampling

Convenience sampling involves selecting the easiest population members from which to obtain information. When the auditor reviews the first 500 student-aid packages he comes across without any random process, it's an instance of convenience sampling.
04

- Describing Cluster Sampling

Cluster sampling is similar to stratified sampling in that it involves dividing the population into clusters, but unlike stratified sampling which requires sampling from every group, cluster sampling involves selecting entire clusters. In this scenario, where schools are selected at random and then majors within those schools, followed by students within those majors, it is a form of cluster sampling.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Census
A census is the most comprehensive form of data collection where every member of the population is included in the study. Unlike sampling methods, there's no selection or exclusion鈥攅very single entity is examined. In the context of the exercise, when the auditor reviews all 15,000 student-aid packages, this is a census because each and every package is scrutinized.

Conducting a census guarantees completeness of data, which is essential in certain situations where precise information is required. However, it's also time-consuming and expensive, which makes sampling methods more appealing in many cases. Interestingly, a census is often seen in governmental studies, such as the decennial census performed in many countries to collect demographic information.
Stratified Sampling
Stratified sampling is a precise way of capturing the diversity within a population. It involves organizing the population into homogeneous subgroups called strata, and then randomly selecting samples from each subgroup.

This ensures that every subgroup is represented in the sample, making the results more generalizable across the population. In the exercise, the auditor demonstrates stratified sampling by choosing 200 student-aid packages from Category 1, 100 from Category 2, and 100 from Category 3. What makes it 'stratified' is the proportionate representation of each category in the sample, mirroring the larger population breakdown.
Convenience Sampling
Convenience sampling is the go-to method when ease and speed are priorities. As its name implies, this method involves taking a sample from a population that is conveniently available.

For example, in the exercise, the auditor's review of the first 500 student-aid packages he comes across is classic convenience sampling. It's quick and effortless, but it does have a significant downside: it's less likely to represent the population accurately, as it may be biased towards certain attributes that make the samples more readily accessible.
Cluster Sampling
Cluster sampling breaks the population down into clusters that are randomly selected to be included in the study. It is often utilized when studying geographically dispersed populations or when a list of all members is unavailable.

In our exercise, the auditor demonstrates cluster sampling by first selecting schools at random, then majors within those schools, and finally students within those majors. In essence, it's a multi-stage selection process where entire mini-populations (clusters) are assessed, rather than individuals from within each stratum as with stratified sampling.
Random Sampling
Random sampling is the gold standard for fairness in selection; each member of the population has an equal and independent chance of being included. This method is known for its ability to minimize selection bias, providing a sample that is representative of the entire population.

While not explicitly presented in the exercise, random sampling is inherent in some of the described methods. For instance, in stratified sampling, after separating the population into strata, members must be chosen at random from each stratum. Similarly, the selection of schools, majors, and students in cluster sampling should also be random to ensure that the sample isn't skewed by a predictable pattern.

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Most popular questions from this chapter

Refer to a landmark study conducted in 1896 in Denmark by Dr. Johannes Fibiger, who went on to receive the Nobel Prize in Medicine in \(1926 .\) The purpose of the study was to determine the effectiveness of a new serum for treating diphtheria, \(a\) common and often deadly respiratory disease in those days. Fibiger conducted his shudy over a one-year period (May 1896 April 1897 ) in one particular Copenhagen hospital. New diphtheria patients admilted to the hospital received different treatments based on the day of admission. In one set of days (call them "even" days for convenience), the patients were treated with the new serum daily and received the standard treatment. Patients admitted on alternate days (the "odd" days) received just the standard treatment. Over the one-year period of the study, eight of the 239 patients admitted on the "even" days and treated with the serum died, whereas 30 of the 245 patients admitted on the "odd" days died. (a) Describe the sample for Fibiger's study. (b) Is selection bias a possible problem in this study? Explain.

Refer to the following story: The dean of students at Tasmania State University wants to determine how many undergraduates at TSU are familiar with a new financial aid program offered by the university. There are 15,000 undergraduates at TSU, so it is too expensive to conduct a census. The following sampling method is used to choose a representative sample of undergraduates to poll. Start with the registrar's alphabetical listing containing the names of all undergraduates. Randomly pick a number between \(l\) and \(100,\) and count that far down the list. Take that name and every I00th name after it. For example, if the random number chosen is \(73,\) then pick the \(73 \mathrm{rd}, 173 \mathrm{rd}, 273 \mathrm{rd}\) and so forth, names on the list. (The sampling method illustrated in this survey is known as systematic sampling.) (a) Explain why the method used for choosing the sample is not simple random sampling. (b) If \(100 \%\) of those responding claimed that they were not familiar with the new financial aid program offered by the university, is this result more likely due to sampling variability or to sample bias? Explain.

Darroch's method. is a method for estimating the size of a population using multiple (more than two) captures. For example, suppose that there are four captures of sizes \(n_{1}, n_{2}, n_{3},\) and \(n_{4},\) respectively, and let \(M\) be the total number of distinct individuals caught in the four captures (i.e., an individual that is captured in more than one capture is counted only once). Darroch's method gives the estimate for \(N\) as the unique solution of the equation \(\left(1-\frac{M}{N}\right)=\left(1-\frac{n_{1}}{N}\right)\left(1-\frac{n_{2}}{N}\right)\left(1-\frac{n_{3}}{N}\right)\left(1-\frac{n_{4}}{N}\right) .\) (a) Suppose that we are estimating the size of a population of fish in a pond using four separate captures. The sizes of the captures are \(n_{1}=30, n_{2}=15, n_{3}=22\), and \(n_{4}=45 .\) The number of distinct fish caught is \(M=75 .\) Estimate the size of the population using Darroch's formula. (b) Show that with just two captures Darroch's method gives the same answer as the capture-recapture method.

Refer to the following story: An orange grow. er wishes to compute the average yield from his orchard. The orchard contains three varieties of trees: \(50 \%\) of his trees are of variety \(A, 25 \%\) of variety \(B,\) and \(25 \%\) of variety \(C\) (a) Suppose that the grower samples randomly from 300 trees of variety A, 150 trees of variety \(B\), and 150 trees of variety C. What type of sampling is being used? (b) Suppose that the grower selects for his sample a 10 by 30 rectangular block of 300 trees of variety \(A\), a 10 by 15 rectangular block of 150 trees of variety \(B\), and a 10 by 15 rectangular block of 150 trees of variety \(\mathrm{C}\). What type of sampling is being used?

Refer to the following story: An orange grow. er wishes to compute the average yield from his orchard. The orchard contains three varieties of trees: \(50 \%\) of his trees are of variety \(A, 25 \%\) of variety \(B,\) and \(25 \%\) of variety \(C\) (a) Suppose that in his survey, the grower found that each tree of variety A averages 100 oranges, each tree of variety \(\mathrm{B}\) averages 50 oranges, and each tree of varietyC averages 70 oranges. Estimate the average yield per tree of his orchard. (b) Is the yield you found in (a) a parameter or a statistic? Explain.
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