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.
Yes or No
b) the Chebyshev bound is too conservative.
Yes or No
c) the people sampled are all different, so their answers are not identically distributed.
Yes or No
d) the people sampled do not have independent opinions.
Yes or No
a). Yes
b). Yes
c). No
d). No
yes,yes,no,no
a) No
b) Yes
c) Yes
d) Yes
a) No. The accuracy requirements being looser is not a valid reason for using smaller sample sizes. The required sample size for a certain level of accuracy is determined by statistical theory and probability calculations, which should be followed to maintain reliable results.
b) Yes. The Chebyshev bound is a very conservative estimation of sample size. In practice, researchers often use more precise and specialized formulas, such as the Central Limit Theorem or the Margin of Error, to determine the appropriate sample size for their specific study. These methods may take into account additional information about the population or the desired level of precision.
c) Yes. If the people being sampled have different characteristics or opinions that are not representative of the population as a whole, then the sample size required to achieve a certain level of accuracy may be smaller. In such cases, using a smaller sample size can still provide useful insights but may not accurately estimate the population as precisely.
d) No. The independence of opinions is not a valid reason for using smaller sample sizes. In fact, if opinions are not independent, it may be even more important to have a larger sample size to capture the complexity of the relationships between variables and to ensure more reliable and accurate results.