Exercise: Polling
4 points possible (graded)
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=500 . 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.
Select an option
unanswered
b) the Chebyshev bound is too conservative.
Select an option
unanswered
c) the people sampled are all different, so their answers are not identically distributed.
Be careful in parts c) and d) (April 17): You are not asked whether the statement is true or false, but whether it would be a valid reason for Newspaper polls to use smaller sample sizes.
Select an option
unanswered
d) the people sampled do not have independent opinions.
Select an option
a) the accuracy requirements are looser.
This statement is a valid reason for Newspaper polls to use smaller sample sizes. In practice, the tolerance for accuracy in poll results may be lower than the 95% suggested by Chebyshev's inequality. Therefore, using smaller sample sizes can still provide sufficient accuracy given the looser requirements.
b) the Chebyshev bound is too conservative.
This statement is a valid reason for Newspaper polls to use smaller sample sizes. Chebyshev's inequality provides a conservative estimation of the sample size needed to achieve a certain level of accuracy. In practice, other statistical methods and techniques may be used that are more efficient and can yield accurate results with smaller sample sizes.
c) the people sampled are all different, so their answers are not identically distributed.
This statement is not a valid reason for Newspaper polls to use smaller sample sizes. Even if the people sampled are different and their answers may not be identically distributed, this does not necessarily result in smaller sample sizes being appropriate. The size of the sample needed for accurate results is influenced by various factors, including the desired level of precision and the variability in the population being sampled.
d) the people sampled do not have independent opinions.
This statement is a valid reason for Newspaper polls to use smaller sample sizes. When the people sampled do not have independent opinions, it means that the correlation or similarity between their opinions can affect the required sample size. If the opinions are strongly correlated, it may be possible to achieve accurate results with smaller sample sizes. However, it is important to consider other factors such as the variability in the population and the desired level of precision when determining an appropriate sample size.