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Chapter 2: Problem 14

A petition with 500 signatures is submitted to a university's student council. The council president would like to determine the proportion of those who signed the petition who are actually registered students at the university. There is not enough time to check all 500 names with the registrar, so the council president decides to select a simple random sample of 30 signatures. Describe how this might be done.

Short Answer

Expert verified
Use a random number generator to choose 30 signatures from the 500 on the petition for verification.

Step by step solution

01

Understanding Simple Random Sample

A simple random sample is a subset of a population in which each member of the subset has an equal probability of being chosen. In this case, the population is the 500 signatures on the petition.
02

Develop a Procedure for Selection

One approach could be to assign each of the 500 signatures a unique number, say from 1 to 500. Then use a random number generator to select 30 distinct numbers from 1 to 500. The signatures corresponding to these numbers constitute the simple random sample that will be checked for university registration.
03

Explain the Process

A random number generator will be used to ensure the fairness and randomness of the selection process. Any signature has the same chance of being included in the sample as any other, which is the defining characteristic of a simple random sample. The 30 selected signatures will be presented to the registrar to confirm if they are registered university students.

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Key Concepts

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

Random Number Generator
When dealing with a large group of items or individuals, such as the 500 signatures on a petition, a random number generator is a valuable tool. This tool helps create a fair and unbiased selection process. Using technology, like a computer algorithm or a calculator feature, random numbers can be generated easily.
This randomness is crucial because it ensures that each item or individual in the list has no prior expectation or chance of being selected based on human choice.
For the petition example, after assigning each signature a unique number from 1 to 500, the generator can select 30 random numbers. Each number corresponds to a signature. The randomness ensures that the sample accurately represents the larger group, free from bias or patterns.
Population and Sample
In statistics, it's important to differentiate between a population and a sample. The population refers to the entire group that you want to draw conclusions about. In our case, it's all 500 signatures on the petition.
A sample, however, is a smaller group selected from the population, used to estimate the characteristics of the entire group. Here, the sample is the 30 signatures chosen from the 500.
  • The sample must be representative to make accurate conclusions about the whole population.
  • The selection should always be random to prevent biased sampling.
By studying the 30 signatures, the council can infer if similar proportions apply to all petition signers, helping them understand the registration status in a faster, efficient way.
Probability
Probability plays a critical role in the process of selecting a sample. It refers to the likelihood or chance that an event will occur. In simple random sampling, each individual or item has an equal probability of being chosen.
For example, if 30 names are selected out of 500, each signature has a probability of \( \frac{1}{500} \). This procedure ensures that the sample is unbiased and gives each person or item in the population the same opportunity of selection, maintaining the integrity of the research task.
Understanding probability helps in calculating the confidence we can have regarding the accuracy of our sample's representation of the population.
Equal Chance Selection
Equal chance selection is a fundamental principle in simple random sampling. It ensures that every member of the population has the same probability of being included in the sample.
  • This is achieved by using unbiased methods like random number generators.
  • It prevents favoritism or systematic preference for certain items or individuals.

In the context of the student council鈥檚 petition, each of the 500 signatures had an equal shot at being picked among the 30 inspected. This fairness in opportunity makes simple random sampling a preferred method in statistical studies, balancing effort with accuracy in analysis. Accurate representation through equal chance selection means conclusions drawn can be trusted to reflect the true characteristics of the entire population.

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Most popular questions from this chapter

For each of the situations described, state whether the sampling procedure is simple random sampling, stratified random sampling, cluster sampling, systematic sampling, or convenience sampling. a. All first-year students at a university are enrolled in 1 of 30 sections of a seminar course. To select a sample of freshmen at this university, a researcher selects four sections of the seminar course at random from the 30 sections and all students in the four selected sections are included in the sample. b. To obtain a sample of students, faculty, and staff at a university, a researcher randomly selects 50 faculty members from a list of faculty, 100 students from a list of students, and 30 staff members from a list of staff. c. A university researcher obtains a sample of students at his university by using the 85 students enrolled in his Psychology 101 class. d. To obtain a sample of the seniors at a particular high school, a researcher writes the name of each senior on a slip of paper, places the slips in a box and mixes them, and then selects 10 slips. The students whose names are on the selected slips of paper are included in the sample. e. To obtain a sample of those attending a basketball game, a researcher selects the 24 th person through the door. Then, every 50 th person after that is also included in the sample.

Consider the following graphical display that appeared in the New York Times: Based on the data summarized in the graph, we can see that students who have a high school GPA or 3.5 or higher and a combined SAT score of over 1200 have an \(89 \%\) graduation rate when they attend a "most selective" college, but only a \(59 \%\) graduation rate when they attend a "least selective" college. Give an example of a potential confounding variable that might explain why the following statement is not reasonable: If all the students that have a GPA of 3.5 or higher and a combined SAT score of 1200 or higher and that were admitted to a "least selective" college were moved to a "most selective" college, the graduation rate for these students would be approximately \(89 \%\)

Briefly explain why it is advisable to avoid the use of convenience samples.

The paper "Deception and Design: The Impact of Communication Technology on Lying Behavior" (Computer-Human Interaction [2009]: \(130-136\) ) describes an investigation into whether lying is less common in face-to-face communication than in other forms of communication such as phone conversations or e-mail. Participants in this study were 30 students in an upperdivision communications course at Cornell University who received course credit for participation. Participants were asked to record all of their social interactions for a week, making note of any lies told. Based on data from these records, the authors of the paper concluded that students lie more often in phone conversations than in face-to-face conversations and more often in face-to-face conversations than in e-mail. Discuss the limitations of this study, commenting on the way the sample was selected and potential sources of bias.

The article "Television's Value to Kids: It's All in How They Use It" (Seattle Times, July 6,2005\()\) described a study in which researchers analyzed standardized test results and television viewing habits of 1700 children. They found that children who averaged more than 2 hours of television viewing per day when they were younger than 3 tended to score lower on measures of reading ability and short-term memory. a. Is the study described an observational study or an experiment? b. Is it reasonable to conclude that watching two or more hours of television is the cause of lower reading scores? Explain.
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