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Chapter 7: Problem 14

Suppose you want to estimate the mean grade point average (GPA) of all students at your school. You set up a table in the library asking for volunteers to tell you their GPAs. Do you think you would get a representative sample? Why or why not?

Short Answer

Expert verified
No, the method of sampling used in the example is not likely to result in a representative sample of all students' GPAs at the school due to the possibility of bias.

Step by step solution

01

Define Representative Sample

A representative sample is a group from a larger population that accurately reflects the characteristics of the whole population.
02

Analyze the Sampling Method

In the given exercise, the method of sampling is voluntary response sampling, which is a type of non-probability sampling. This method invites all individuals from a population to participate, and those who choose to participate are included in the sample.
03

Identify Possibility of Bias

Bias might occur when not all segment of population are equally likely to participate. Students who hang out at the library might be more academically inclined than those who don't, and hence GPAs reported might tend to be higher. Also, students with lower GPAs might not feel comfortable sharing.
04

Conclusion

Bias, observed in Step 3, suggests that this method may not result in a representative sample of all students at the school. It's likely that the sample will not adequately represent students with lower GPAs, students who do not frequent the library or those too shy about their performance.

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Key Concepts

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

Representative Sample
A representative sample serves as a miniature version of the larger population from which it is drawn. It should accurately mirror the diverse characteristics and variations found within the entire group.
This could include demographics, behaviors, or in the context of the exercise, academic performance like GPA.
A well-constructed representative sample enables researchers to make reliable and valid conclusions about the whole population without needing to survey everyone.
  • For a sample to be truly representative, every individual in the population must have an equal chance of being selected.
  • If certain groups are over- or under-represented, the sample may produce misleading results.
Ensuring that the sample is representative minimizes bias and enhances the accuracy of research findings.
Voluntary Response Sampling
Voluntary response sampling relies on participants who choose to be involved, rather than being randomly selected.
It's a straightforward method to implement, but it often comes with limitations in terms of validity and reliability.
  • This method tends to attract individuals with strong opinions or certain traits, which are not necessarily typical of the general population.
  • In the context of the GPA exercise, students visiting the library are more likely to volunteer if they are confident in their GPAs, potentially skewing the results upwards.
Despite its ease of use, voluntary response sampling can significantly reduce the representativeness of a sample, as it misses out on the diverse range of perspectives and experiences found within a complete population.
Bias in Sampling
Bias in sampling occurs when the sample selected deviates from the population it aims to represent, introducing systematic error.
This can result from flawed sampling methods or external influences affecting participation.
  • In the GPA estimation exercise, bias arises because certain students are either more likely to participate or are actively excluded due to the nature of the sampling method.
  • For instance, students with lower GPAs might avoid participating out of discomfort, while high-achieving students might be more willing to share their GPAs.
Acknowledging and addressing potential bias is crucial in research design, as it ensures the findings are accurate and reflective of the true population characteristics. Adjustments or alternative methods, like random sampling, can be employed to mitigate bias and achieve more balanced results.

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Most popular questions from this chapter

A poll on a proposition showed that we are \(95 \%\) confident that the population proportion of voters supporting it is between \(40 \%\) and \(48 \%\). Find the margin of error.

Find the sample size required for a margin of error of 3 percentage points, and then find one for a margin of error of \(1.5\) percentage points; for both, use a \(95 \%\) confidence level. Find the ratio of the larger sample size to the smaller sample size. To reduce the margin of error to half, by what do you need to multiply the sample size?

The Centers for Disease Control and Prevention (CDC) conducts an annual Youth Risk Behavior Survey, surveying over 15,000 high school students. The 2015 survey reported that, while cigarette use among high school youth had declined to its lowest levels, \(24 \%\) of those surveyed reported using e-cigarettes. Identify the sample and population. Is the value \(24 \%\) a parameter or a statistic? What symbol would we use for this value?

In the 2018 study Closing the STEM Gap, researchers wanted to estimate the percentage of middle school girls who planned to major in a STEM field. a. If a \(95 \%\) confidence level is used, how many people should be included in the survey if the researchers wanted to have a margin of error of \(3 \%\) ? b. How could the researchers adjust their margin of error if they want to decrease the number of study participants?

The mean weight of all professional NBA basketball players is \(218.8\) pounds. A sample of 50 professional basketball players has a mean weight of \(217.6\) pounds. Which number is \(\mu\), and which number is \(\bar{x}\) ?
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