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Chapter 1: Problem 36

In Exercises 1.36 to 1.39 , a biased sampling situation is described. In each case, give: (a) The sample (b) The population of interest (c) A population we can generalize to given the sample To estimate the proportion of Americans who support changing the drinking age from 21 to \(18,\) a random sample of 100 college students are asked the question "Would you support a measure to lower the drinking age from 21 to \(18 ?\) "

Short Answer

Expert verified
Sample: 100 college students; Population of interest: All Americans; Population we can generalize to: College students in the U.S.

Step by step solution

01

Identify the sample

The sample in this scenario is the 100 college students that were randomly selected for the survey.
02

Identify the population of interest

The population of interest, in this case, represent all Americans, since the purpose of the survey is to estimate the proportion of Americans who support changing the drinking age from 21 to 18.
03

Identify a population we can generalize to given the sample

While the population of interest is all Americans, it's important to remember that the sample for this study is made up of college students. Because of this, the survey findings are more representative of college students' opinions, and therefore, we can generalize this to the population of college students in the U.S.

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Key Concepts

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

Sample
In the context of statistics, a "sample" refers to a smaller group selected from a larger population, which is the bigger group we are interested to learn about.
In our exercise, the sample consists of the 100 college students who were randomly selected to participate in the survey. These students represent a portion of a larger group.

A good sample should ideally reflect the composition of the whole population as much as possible. In this case, however, only college students are asked about their opinion on changing the drinking age. This raises questions about whether their views truly reflect those of the broader group of Americans.
  • **Random Selection**: Ideally helps in achieving a diverse sample but does not always mirror the overall population's diversity.
  • **Size Matters**: Larger samples tend to provide more reliable and stable insights about the population.
  • **Bias Risk**: When a sample is not representative, like here, it may lead to misleading conclusions about the whole population.
Population of Interest
The term "population of interest" signifies the entire group of individuals that we want to draw conclusions about. In this exercise, that's all Americans.
This is because the researchers are eager to understand how Americans feel about the proposed change in the legal drinking age from 21 to 18.

The idea is to make an inference about this large group's opinions by surveying a smaller portion of them, focusing on specific questions.
  • **Scope and Relevance**: It's not solely about the size but about identifying which group's opinions are pertinent to the study's demands.
  • **General Goals**: The aim here is to gauge the support for lowering the drinking age in the entire American populace.
  • **Realistic Constraints**: Sometimes, it's impractical to survey everyone, hence the need for a representative sample.
This population might include various demographics, and it is crucial to use appropriate sampling techniques to attempt guiding broad and applicable results.
Generalization in Statistics
Generalization in statistics is the process of applying findings from a sample to a broader population.
It involves predicting how the entire population will respond based on the sample's data.
In the exercise at hand, the researchers intended to generalize their findings from the college students' sample to represent the opinions of all Americans. However, due to the sampling bias, this generalization is not fully justified.

Here are some points to understand generalization better:
  • **Representativeness**: The sample must accurately reflect the larger population's characteristics. If not, the generalizations made could be skewed or biased.
  • **Limitations**: A sample of college students may not have the same views as the entire American population, given differences in experiences, age, and social factors.
  • **Improving Accuracy**: By broadening the sample to include a more diverse group of participants, including non-students, a more accurate generalization might be achieved.
This understanding highlights why choosing the right sample is critical for making accurate and meaningful inferences about a broader group.

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Most popular questions from this chapter

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