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A survey of affluent Americans (those with incomes of $$ 75,000\( or more) indicated that \)57 \%$ would rather have more time than more money (USA Today, January 29, 2003). a. What condition on how the data were collected would make it reasonable to generalize this result to the population of affluent Americans? b. Would it be reasonable to generalize this result to the population of all Americans? Explain why or why not.

Short Answer

Expert verified
Conditions that would make it reasonable to generalize this result to the population of affluent Americans include if a random and representative sample of the group was used in the survey, and if the sample size was statistically significant. However, it would not be reasonable to generalize this result to the population of all Americans without survey data from other income groups, as their circumstances, preferences, and characteristics may be significantly different.

Step by step solution

01

Analysis of Data Collection Condition

A condition that would make it reasonable to generalize this result to the population of affluent Americans is if the data were obtained from a random and representative sample of this group. The sample should be large enough to be statistically significant, meaning the sample size should be sufficient to confidently estimate the behavior or opinion of the entire affluent American population.
02

Assessment of Generalization to All Americans

Generalizing this result to the population of all Americans would not be reasonable. The original survey was conducted only among affluent Americans, those with incomes of $75,000 or more. However, the characteristics, preferences, and circumstances of these individuals may be significantly different from those of all Americans, including those in other income brackets. Therefore, without survey data from these other groups, it would not be appropriate to generalize this result to all Americans.
03

Summary of Reasoning

In statistical analysis, making valid generalizations requires careful consideration of the population of interest and the conditions under which the data were collected. If the data were collected from representative and random samples of a specific group, such as affluent Americans, it could be reasonable to generalize these results to that specific group. However, generalizing findings to a broader or different population (e.g., all Americans) requires additional data and sound reasoning, to ensure the generalization is appropriate and reliable.

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Key Concepts

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

Random Sampling
In order to draw accurate conclusions from a survey, it is crucial to utilize random sampling. This method involves selecting participants in such a way that every individual in the population has an equal chance of being chosen. This equality in chance ensures that the sample is free from bias, allowing the survey results to be reflective of the larger population.

Random sampling is particularly important in ensuring that the results are not skewed by over-representing or under-representing specific subgroups. Without randomness, the outcomes could be misleading, as they might highlight the tendencies of overrepresented groups rather than the entire population's characteristics.
  • Reduces bias and ensures diversity
  • Makes findings more credible and reliable
  • Facilitates the application of statistical techniques
For example, if a survey about affluent Americans uses random sampling, every affluent individual would hypothetically have the same opportunity to participate, lending credibility to the survey's generalizability to the broader group.
Representative Sample
A representative sample is critical for drawing conclusions about a larger population. It means that the sample reflects the various traits and characteristics of the overall group from which it is drawn. For a survey to produce useful insights, the sample should embody diversity regarding age, income, education, and other relevant characteristics of the population.

In the context of a survey about affluent Americans, a representative sample would not only include a variety of individuals meeting the income bracket criteria, but also account for geographical distribution, occupational fields, and family sizes—among other potential factors.
  • Mirrors the population's diversity
  • Improves the accuracy of survey conclusions
  • Essential for valid generalization
This ensures that the survey outcomes truly reflect the perspectives and preferences of the entire group, allowing educators, policymakers, or businesses to make informed decisions based on the survey data.
Statistical Significance
Statistical significance plays a central role in interpreting survey results. It helps determine if the findings of a survey truly reflect the population or if they might have occurred by random chance. When results are statistically significant, it implies that the observed effects are likely genuine, and not merely the result of random fluctuations.

To establish statistical significance, sample size and proper sampling methods are vital. A survey about affluent Americans must involve enough participants to ensure that any observed trend or characteristic would likely appear if the entire population was surveyed.
  • Validates the reliability of survey findings
  • Indicates confidence in generalizing results
  • Requires adequate sample size
In scenarios lacking statistical significance, the findings could be dismissed as unreliable. Therefore, calculating and demonstrating statistical significance is essential for turning raw survey data into actionable insights that confidently represent a larger group's opinions or behaviors.

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Most popular questions from this chapter

For each of the statistical studies described indicate whether the study is an observational study or an experiment. Give a brief explanation for your choice. The following conclusion from a statistical study appeared in the article "Smartphone Nation" (AARP Bulletin, September 2009): "If you love your smart phone, you are not alone. Half of all boomers sleep with their cell phone within arm's length. Two of three people ages 50 to 64 use a cell phone to take photos, according to a 2010 Pew Research Center report."

The article "Super Bowls: Serving Bowl Size and Food Consumption" (Journal of the American Medical Association [2005]: 1727-1728) describes an experiment investigating how the size of serving bowls influences the amount a person eats. In this experiment, graduate students at a university were recruited to attend a Super Bowl party. The paper states that as the students arrived, they were "led in an alternating order to one of two identical buffet tables on opposite sides of an adjoining room. The tables had identical amounts of snacks, such as nuts, pretzels, and chips. All of the snacks contained approximately the same number of calories per gram. On one of the tables the snacks were set out in large serving bowls and on the second table the snacks were set out in smaller serving bowls. The students were given a plate and invited to serve themselves before going to another room to watch the game. When they arrived at the game room, their plates were weighed and the number of calories in the food on the plate was estimated." The researchers concluded that serving bowl size does make a difference, with those using large serving bowls tending to take more food. a. Do you think that the alternate assignment to the experimental groups (large serving bowls, small serving bowls) based on arrival time is "close enough" to random assignment? That is, do you think it would tend to create comparable experimental groups? b. In this study, the research assistant who weighed the plates and estimated the calorie content of the food on the plate was blinded as to which experimental group the plate belonged to and was also blinded as to the purpose of the experiment. Why do you think the researchers chose to incorporate this type of blinding?

For each of the statistical studies described, indicate whether the study is an observational study or an experiment. Give a brief explanation for your choice. In a study of whether taking a garlic supplement reduces the risk of getting a cold, 146 participants were assigned to either a garlic supplement group or to a group that did not take a garlic supplement ("Garlic for the Common Cold," Cochrane Database of Systematic Reviews, 2009). Based on the study, it was concluded that the proportion of people taking a garlic supplement who get a cold is lower than the proportion of those not taking a garlic supplement who get a cold.

The same article referenced in Exercise 1.13 also described a study which concluded that people tend to respond differently to the following questions: Question 1: Would you rather have \(\$ 50\) today or \(\$ 52\) in a week? Question 2 : Imagine that you could have \(\$ 52\) in a week. Would you rather have \(\$ 50\) now?

Suppose that you would like to know if keyboard design has an effect on wrist angle, as shown in the accompanying figure. You have 40 volunteers who have agreed to participate in an experiment to compare two different keyboards. Describe a method for randomly assigning each of the volunteers to one of the experimental groups.

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