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Chapter 14: Problem 23

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

Short Answer

Expert verified
The population for the survey is all the 1250 students at Eureka High School while the sampling frame is the subset of students who attended the game. The difference between population and sampling frame could result in a sampling bias if not all kinds of students were equally likely to attend the football game. The sampling error in this context is likely to comprise of both - sampling variability inherent in the process of sampling, and bias if the sample is not representative of the population.

Step by step solution

01

Identify the population and sample

The population for this survey is all the 1250 students at Eureka High School. The sample for this survey is the group of students who attended the football game and participated in the pre-election poll.
02

Compare and contrast population and sampling frame

The population refers to the whole group of 1250 students. On the other hand, the sampling frame, that is the group of students from which the sample is drawn, is the students who attended the football game. Thus, while the population includes all students, the sampling frame is a subset comprising of students who attended a particular event. It is possible that the sampling frame might not fully represent the population if, for example, not all kinds of students (in terms of their preferences, affiliations, etc.) were equally likely to attend the football game.
03

Identify sampling error and determine whether it is due to variability or bias

Sampling error is the difference between a statistic computed from a sample and the true population statistic. This error can result from two sources - sampling variability and bias. Sampling variability is due to the fact that different samples would likely yield different statistics; it is inherent in the process of sampling and decreases as sample size increases. Bias, on the other hand, is a systematic error that could occur if the sample is not representative of population. Considering the context, if the students who attended the football game (and hence participated in the pre-election poll) are not a diverse and representative set of the student body, there could be sampling bias. For example, if more football enthusiasts (who are more likely to support Tomlinson, the football team captain) attend the game, Tomlinson may get a higher number of votes in the pre-election poll than he would in the actual election poll across all students.

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Key Concepts

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

Population and Sample
When we talk about population and sample, we're referring to two key components of any research study. In our story from Eureka High School:
  • The population is all 1250 students who could potentially vote in the Homecoming King election. This entire group is the focus of the study.
  • The sample is the group of students who attended the football game and participated in a pre-election poll. They represent only a portion of the total population.
The purpose of sampling is to gather insights that could apply to the whole population without having to survey every single student. However, the effectiveness of this approach depends heavily on how well the sample reflects the population as a whole.
If the sample is not representative, the results can be misleading. It鈥檚 like trying to guess the flavor of a whole cake by tasting just one slice 鈥� if that slice doesn鈥檛 have a good spread of ingredients, your guess might not be accurate.
Sampling Frame
The sampling frame is crucial in understanding how a sample is selected. It's essentially a list or group from which a sample is drawn. In the case of Eureka High School:
  • The sampling frame is the group of students who attended the football game.
This group makes up the pool from which the sample can be chosen. It鈥檚 important to note that a sampling frame doesn鈥檛 always perfectly match the population. Sometimes the frame can be limited by practical constraints, like only having access to certain groups at specific times. In the Eureka High School example, only students attending the game were included in the poll.
This might leave out those with different interests or commitments, therefore impacting how well the sampling frame represents the entire student body.
Sampling Variability
Sampling variability describes the natural fluctuations that occur when different samples are taken from the same population. Imagine taking multiple groups of students from Eureka High School at different events.
  • Each sample might give slightly different results because the students attending each event may vary.
  • With a larger sample size, these differences, or variability, tend to be smaller, leading to more stable results.
However, the ideal sample reflects the diversity and characteristics of the entire population. If sampling variability is high, the results can differ significantly from sample to sample, making it challenging to draw reliable conclusions about the population as a whole. It's a key reason why researchers aim for large and randomized samples, to minimize variability and increase accuracy.
Sample Bias
Sample bias occurs when a sample does not accurately reflect the population. It can lead to skewed results because certain groups are overrepresented while others are underrepresented. In Eureka High School鈥檚 context:
  • If mostly football enthusiasts participated in the pre-election poll, their preferences could skew the results.
  • This bias might make it look like Tomlinson is more popular than he actually is among all students.*
Sample bias can happen for a number of reasons, such as when the sampling frame is too narrow or certain groups have unequal chances of being selected. Recognizing and minimizing sample bias is crucial for making sure that conclusions drawn from the sample apply to the entire population, not just a subset with specific characteristics.

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Most popular questions from this chapter

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 a study conducted between 2008 and 2010 on the effectiveness of saw palmetto fruit extracts at treating lower urinary tract symptoms in men with prostate enlargement. (Saw palmetto is a widely used over-the-counter supplement for treating urinary tract symptoms.) In the study, 369 men aged 45 years or older were randomly divided into a group taking a daily placebo and a group taking saw palmetto. Participants were nonpaid volunteers recruited at 11 North American sites. All had moderately impaired urinary flow. Because the saw palmetto extract has a mild odor, the doses were administered using gelcaps to eliminate the odor. In an analysis of the 306 men who completed the 72 -week trial, both groups had similar small improvements in mean symptom scores, but saw palmetto conferred no benefit over placebo on symptom scores or on any secondary outcomes. (a) Describe the treatment group in the study. (b) Explain why the experimenters took the trouble to cover the mild odor of saw palmetto to the point of packaging the doses in the form of gelcaps. (c) Was this study a blind, randomized, controlled placebo study? Explain.

Today, most consumer marketing surveys are conducted by telephone. In selecting a sample of households that are representative of all the households in a given geographical area, the two basic techniques used are (1) randomly selecting telephone numbers to call from the local telephone directory or directories and (2) using a computer to randomly generate seven-digit numbers to try that are compatible with the local phone numbers. (a) Briefly discuss the advantages and disadvantages of each technique. In your opinion, which of the two will produce the more reliable data? Explain. (b) Suppose that you are trying to market burglar alarms in New York City. Which of the two techniques for selecting the sample would you use? Explain your reasons.

Refer to a clinical trial named APPROVe designed to determine whether Vioxx, a medication used for \(a r\) thritis and acute pain, was effective in preventing the recurrence of colorectal polyps in patients with a history of colorectal adenomas. APPROVe was conducted between 2002 and 2003 and involved 2586 participants, all of whom had a history of colorectal adenomas. The participants were randomly divided into two groups: 1287 were given 25 milligrams of Vioxx daily for the duration of the clinical trial (originally intended to last three years), and 1299 patients were given a placebo. Neither the participants nor the doctors involved in the clinical trial knew who was in which group. During the trial, 72 of the participants had cardiovascular events (mostly heart attacks or strokes). Later it was found that 46 of these people were from the group taking the Vioxx and only 26 were from the group taking the placebo. Based on these results, the clinical trial was stopped in 2003 and Vioxx was taken off the market in 2004. Describe the sample for APPROVe.

Refer to a study on the effectiveness of an HPV (human papilloma virus) vaccine conducted between October 1998 and November \(1999 .\) HPV is the most common sexually transmitted infection-more than 20 million Americans are infected with HPV-but most HPV infections are benign, and in most cases infected individuals are not even aware they are infected. (On the other hand, some HPV infections can lead to cervical cancer in women.) The researchers recruited 2392 women from 16 different centers across the United States to participate in the study through advertisements on college campuses and in the surrounding communities. To be eligible to participate in the study, the subjects had to meet the following criteria: (1) be a female between 16 and 23 years of age, (2) not be pregnant, (3) have no prior abnormal Pap smears, and (4) report to have had sexual relations with no more than five men. At each center, half of the participants were randomly selected to receive the HPV vaccine, and the other half received a placebo injection. After 17.4 months, the incidence of HPV infection was 3.8 per 100 woman-years at risk in the placebo group and 0 per 100 woman-years at risk in the vaccine group. In addition, all nine cases of HPV-related cervical precancerous growths occurred among the placebo recipients. (a) Describe the treatment group in the study. (b) Could this study be considered a double-blind, randomized controlled placebo study? Explain.
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