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Chapter 14: Problem 17

Refer to the following story: The city of Cleansburg has 8325 registered voters. There is an election for mayor of Cleansburg, and there are three candidates for the position: Smith, Jones, and Brown. The day before the election a telephone poll of 680 randomly chosen registered voters produced the following results: 306 people surveyed indicated that they would vote for Smith, 272 indicated that they would vote for Jones and 102 indicated that they would vote for Brown. (a) Describe the population for this survey. (b) Describe the sample for this survey. (c) Name the sampling method used for this survey.

Short Answer

Expert verified
The population for this survey is the 8325 registered voters of Cleansburg. The sample for this survey is the 680 registered voters that were randomly selected and polled. The sampling method used for this survey is simple random sampling.

Step by step solution

01

Identify the Population

The population in this scenario encompasses the total group that is the subject of the study. Here, the population refers to the total registered voters in the city of Cleansburg, which is 8325.
02

Identify the Sample

The sample refers to a subset of the population chosen to participate in the study. In this case, the sample is the 680 randomly chosen registered voters who were polled.
03

Identify the Sampling Method

The sampling method refers to the technique used to select individuals from the population for the study. In this case, randomly selecting 680 registered voters indicates a simple random sampling method, where every member of the population has an equal chance of being selected.

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Key Concepts

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

Population
In statistics, the term "population" refers to the entire group that you want to gather information about or make conclusions on. It encompasses all the members of a defined set. When conducting research, clearly defining the population is essential, as it helps in setting the scope of the study.
In our example from Cleansburg, the population is all 8325 registered voters in the city.
  • This includes every individual who has a say in the election, and thus, their opinions are crucial for gaining accurate insights about the future mayor's support in the city.
  • It's important to note that the population is defined not just by number, but by specific characteristics relevant to the research, such as eligibility to vote in this case.
By understanding the population, researchers can better strategize how to collect relevant data and ensure that their findings are as close to reality as possible.
Sample
A "sample" is essentially a smaller group selected from the population to participate in the study. This group is used to draw conclusions about the entire population due to the infeasibility of studying everyone.
In Cleansburg's scenario, the sample consists of 680 registered voters.
  • Sampling allows researchers to focus on a manageable number of subjects while still attaining results that, if the sample is representative, can reasonably be generalized to the whole population.
  • The goal is to select a sample that mirrors the population, which in this case means that the views of these 680 voters should reflect the opinions of all 8325 voters as closely as possible.
Effective sampling, like in this case, ensures that the insights gained are informative and reliable, allowing researchers to make educated predictions and decisions.
Simple Random Sampling
"Simple random sampling" is a fundamental technique in statistics where each member of the population has an equal opportunity to be included in the sample. It's frequently used because of its straightforwardness and propensity to produce a balanced and unbiased representation of the population.
In Cleansburg's mayoral survey, this method was used to select the 680 participants.
  • Every registered voter, in theory, has an identical chance of being chosen, which helps eliminate selection bias and maintain the integrity of the poll results.
  • This method requires a clear path for interaction, such as a random number generator, ensuring each sample member is picked fairly.
While simple random sampling is effective, it relies on having complete access to the entire population list, which can sometimes be a practical limitation. Nevertheless, it remains a vital tool for achieving dependable data in research.

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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 study over a one-year period (May 1896-April 1897) in one particular Copenhagen hospital. New diphtheria patients admitted 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 as specifically as you can the target population for Fibiger's study. (b) Describe the sampling frame for the study.

Refer to the following story: The 1250 students at Eureka High School are having an election for Homecoming King. The candidates are Tomlinson (captain of the football team), Garcia (class president), and Marsalis (member of the marching band). At the football game a week before the election, a pre- election poll was taken of students as they entered the stadium gates. Of the students who attended the game, 203 planned to vote for Tomlinson, 42 planned to vote for Garcia, and 105 planned to vote for Marsalis. (a) Compare and contrast the population and the sampling frame for this survey. (b) Is the sampling error a result of sampling variability or of sample bias? Explain

Refer to the following story (see also Exercise 32): 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 \(T S U,\) so it is too expensive to conduct a census. The following sampling method, known as systematic sampling, 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 1 and \(100,\) and count that far down the list. Take that name and every 100 th 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. (a) Find the sampling proportion. (b) Suppose that the survey had a response rate of \(90 \%\). Find the size \(n\) of the sample.

Refer to the following story: The manufacturer of a new vitamin (vitamin \(X\) ) decides to sponsor a study to determine the vitamin's effectiveness in curing the common cold. Five hundred college students having a cold were recruited from colleges in the San Diego area and were paid to participate as subjects in this study. The subjects were each given two tablets of vitamin \(X\) a day. Based on information provided by the subjects themselves, 457 of the 500 subjects were cured of their colds within 3 days. (The average number of days a cold lasts is 4.87 days.) As a result of this study, the manufacturer launched an advertising campaign based on the claim that "vitamin \(X\) is more than \(90 \%\) effective in curing the common cold." (a) Was the study a controlled study? Explain. (b) List four possible causes other than the effectiveness of vitamin X itself that could have confounded the results of the study.

(a) For the capture-recapture method to give a reasonable estimate of \(N\), what assumptions about the two samples must be true? (b) Give reasons why the assumptions in (a) may not hold true in many situations.
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