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Chapter 1: Problem 38

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 investigate interest across all residents of the US in a new type of ice skate, a random sample of 1500 people in Minnesota are asked about their interest in the product.

Short Answer

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(a) Sample: 1500 people in Minnesota. (b) Population of interest: All residents of the US. (c) A population we can generalize to given the sample: Residents of US states with similar climates and cultures as Minnesota.

Step by step solution

01

Identify the Sample

In this case, the sample consists of the 1500 people in Minnesota who are asked about their interest in the new ice skate product. These are the individuals from whom data is collected for the study.
02

Identify the Population of Interest

The population of interest here would be all residents of the US. This is because the goal of the investigation is to gauge interest in the ice skate across all residents of the US.
03

Identify a Population to Generalize to Given the Sample

In this case study, the population that can be generalized to given the sample is not the entire US population, because the sample is not representative of the entire US. It is biased because it only includes people from Minnesota, which has a very specific climate (cold weather and ice). The population that the findings can be generalized to could be seen as residents of US states with similar climate and culture as Minnesota.

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Key Concepts

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

Sampling Bias
Understanding sampling bias is crucial when conducting research, as it refers to a scenario where the sample selected is not representative of the entire population. This means that the method of collecting the sample has introduced a kind of non-randomness, where certain group members are systematically excluded, over-represented, or under-represented.

For instance, let's consider the exercise where a sample of 1500 people from Minnesota was taken to represent the US population's interest in a new type of ice skate. The over-representation of Minnesotans in the sample represents a sampling bias, as Minnesotans may have different interests in ice skating than people in warmer states due to the local climate and cultural factors.
Population of Interest
The population of interest is the entire group about which the researcher wants to draw conclusions. It's the larger group from which the sample is drawn and ideally should reflect the sample. In the given exercise, the population of interest is all residents of the United States.

It's essential to clearly define the population of interest in any study to ensure that conclusions drawn are relevant and applicable to that group. If the population of interest differs from the sample, then the findings may not accurately reflect the views or characteristics of the intended group.
Sample Representativeness
A representative sample is one that mirrors the characteristics of the population of interest as closely as possible. Sample representativeness is achieved by using random sampling techniques that give all members of the population an equal chance of being selected.

In the scenario of the 1500 Minnesotans, the representativeness of the sample is questionable as it does not accurately encompass the diversity of climates and cultures across the entire US. Therefore, the sample may not reflect the overall US population's attitude toward the ice skate product.
Generalizing Study Findings
Generalizing study findings to a broader population is the end goal of most research. It involves applying the conclusions drawn from the sample to the population of interest. However, this can only be validly done if the sample that the data is collected from is representative.

In our textbook case, while we can collect data from the Minnesotan sample, the findings can only be generalized to populations that share similar characteristics 鈥� such as those residing in US states with similar winter climates and cultures. Caution must be taken before applying these results across all states, as it may lead to inaccurate assumptions about the entire US population's interest in the new ice skate product.

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Most popular questions from this chapter

For the situations described. (a) What are the cases? (b) What is the variable and is it quantitative or categorical? People in a city are asked if they support a new recycling law.

A study published in 2010 showed that city dwellers have a \(21 \%\) higher risk of developing anxiety disorders and a \(39 \%\) higher risk of developing mood disorders than those who live in the country. A follow-up study published in 2011 used brain scans of city dwellers and country dwellers as they took a difficult math test. \(^{61}\) To increase the stress of the participants, those conducting the study tried to humiliate the participants by telling them how poorly they were doing on the test. The brain scans showed very different levels of activity in stress centers of the brain, with the urban dwellers having greater brain activity than rural dwellers in areas that react to stress. (a) Is the 2010 study an experiment or an observational study? (b) Can we conclude from the 2010 study that living in a city increases a person's likelihood of developing an anxiety disorder or mood disorder? (c) Is the 2011 study an experiment or an observational study? (d) In the 2011 study, what is the explanatory variable and what is the response variable? Indicate whether each is categorical or quantitative. (e) Can we conclude from the 2011 study that living in a city increases activity in stress centers of the brain when a person is under stress?

In Data 1.4 on page \(24,\) we describe the results of a question asked by a national newspaper columnist: "If you had it to do over again, would you have children?" In addition to those results and a follow- up national survey, the Kansas City Star selected a random sample of parents from Kansas City and asked them the same question. In this sample, \(94 \%\) said 鈥淵es." To what population can this statistic be generalized?

Do Cat Videos Improve Mood? As part of an "internet cat videos/photos" study, Dr. Jessica Gall Myrick posted an on-line survey to Facebook and Twitter asking a series of questions regarding how individuals felt before and after the last time they watched a cat video on the Internet. \(^{31}\) One of the goals of the study was to determine how watching cat videos affects an individual's energy and emotional state. People were asked to share the link, and everyone who clicked the link and completed the survey was included in the sample. More than 6,000 individuals completed the survey, and the study found that after watching a cat video people generally reported more energy, fewer negative emotions, and more positive emotions. (a) Would this be considered a simple random sample from a target population? Why or why not? (b) Ignoring sampling bias, what other ways could bias have been introduced into this study?

A study \(^{60}\) examined the impact of the color red on how attractive men perceive women to be. In the study, men were randomly divided into two groups and were asked to rate the attractiveness of women on a scale of 1 (not at all attractive) to 9 (extremely attractive). One group of men were shown pictures of women on a white background and the other group were shown the same pictures of women on a red background. The men who saw women on the red background rated them as more attractive. All participants and those showing the pictures and collecting the data were not aware of the purpose of the study. (a) Is this an experiment or an observational study? Explain. (b) What is the explanatory variable and what is the response variable? Identify each as categorical or quantitative. (c) How was randomization used in this experiment? How was blinding used? (d) Can we conclude that using a red background color instead of white increases men's attractiveness rating of women's pictures?
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