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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

Expert verified
(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

"Antibiotics in infancy may cause obesity in adults," claims a recent headline. \(^{49}\) A study in mice randomly assigned infant mice to either be given antibiotics or not, and the mice given antibiotics were more likely to be obese as adults. A separate study in humans found that children who had been given antibiotics before they were a year old (for example, for an ear infection) were more likely to be obese as adults. (Researchers believe the effect may be due to changes in the gut microbiome.) Based on these studies, is the headline an appropriate conclusion to make: (a) For mice? (b) For humans?

For the situations described. (a) What are the cases? (b) What is the variable and is it quantitative or categorical? Collect data from a sample of teenagers with a question that asks "Do you eat at least five servings a day of fruits and vegetables?"

Describe an association between two variables. Give a confounding variable that may help to account for this association. People who own a yacht are more likely to buy a sports car.

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?

Seven of the ten largest cities in the world are in the Eastern Hemisphere (including the largest: Tokyo, Japan) and three are in the Western Hemisphere. \(^{11}\) Table 1.4 shows the populations, in millions of people, for these cities. (a) How many cases are there in this dataset? How many variables are there and what are they? Is each categorical or quantitative? (b) Display the information in Table 1.4 as a dataset with cases as rows and variables as columns. $$ \begin{array}{ll} \hline \text { Eastern hemisphere: } & 37,26,23,22,21,21,21 \\ \text { Western hemisphere: } & 21,20,19 \\ \hline \end{array} $$
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