We saw that if we want to have a probability of at least 95% that the poll results are within 1 percentage point of the truth, Chebyshev's inequality recommends a sample size of n=50,000 . This is very large compared to what is done in practice. Newspaper polls use smaller sample sizes for various reasons. For each of the following, decide whether it is a valid reason.
In the real world,
a) the accuracy requirements are looser.
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b) the Chebyshev bound is too conservative.
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c) the people sampled are all different, so their answers are not identically distributed.
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d) the people sampled do not have independent opinions.
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Yes
Yes
No
No
a) the accuracy requirements are looser.
Sure, why not? In the real world, we often settle for less than perfection. Maybe we're just not as picky as those statisticians!
b) the Chebyshev bound is too conservative.
Yes, absolutely! Chebyshev is like that one friend who always goes overboard with caution. Sometimes we just want to live life on the edge and take some risks.
c) the people sampled are all different, so their answers are not identically distributed.
Of course! Who wants an identical distribution anyway? It's much more exciting when everyone has their own unique quirks and preferences. Variety is the spice of life!
d) the people sampled do not have independent opinions.
Absolutely! It's a bit unrealistic to assume that everyone thinks independently. People are influenced by their surroundings, experiences, and even their favorite TV shows. We can't expect them to be completely independent and unaffected by external factors.
a) the accuracy requirements are looser - This is a valid reason. In some cases, the required level of accuracy may be lower than 1 percentage point, so a smaller sample size may be sufficient to meet the desired level of accuracy.
b) the Chebyshev bound is too conservative - This is a valid reason. Chebyshev's inequality provides a conservative estimate for the sample size required to achieve a certain level of accuracy. In practice, other statistical methods and bounds may be used, which could result in smaller sample sizes.
c) the people sampled are all different, so their answers are not identically distributed - This is not a valid reason. While it is true that individuals may have different opinions, in statistical sampling, the assumption is that the sampled population is representative of the larger population. Therefore, it is still possible to estimate population parameters with a smaller sample size.
d) the people sampled do not have independent opinions - This is a valid reason. If the opinions of the sampled individuals are highly correlated or influenced by external factors, it may be possible to achieve a desired level of accuracy with a smaller sample size. This is because the correlation or influence among individuals reduces the variability in their responses.
a) the accuracy requirements are looser.
This is a valid reason in the real world. In practice, the accuracy requirements for polling may not need to be as strict as a 95% probability of the poll results being within 1 percentage point of the truth. Depending on the specific circumstances and objectives of the poll, looser accuracy requirements may be sufficient.
b) the Chebyshev bound is too conservative.
This is a valid reason in the real world. Chebyshev's inequality provides a conservative upper bound on the probability that a random variable deviates from its mean by a certain amount. In the context of polling, using Chebyshev's inequality would result in large sample sizes, which may not be necessary or practical in real-world applications. Other methods, such as confidence intervals or hypothesis testing, may provide more appropriate and efficient ways to assess the accuracy of poll results.
c) the people sampled are all different, so their answers are not identically distributed.
This is a valid reason in the real world. One assumption underlying many statistical methods, including those used in polling, is that the random variables being studied are identically distributed. However, in practice, individuals may have different backgrounds, characteristics, or perspectives that could influence their opinions. This heterogeneity among respondents introduces potential bias or variability that may need to be considered when designing and analyzing polls.
d) the people sampled do not have independent opinions.
This is a valid reason in the real world. Many statistical methods, including polling, assume that the observations or individuals being studied have independent opinions or responses. However, in reality, people's opinions can be influenced by various factors, such as social networks, media, or group dynamics. When sampling individuals with interdependent opinions, more sophisticated techniques, such as network sampling or multilevel modeling, may be needed to account for these dependencies and obtain accurate results.