Solutions to Selected Exercises 177formed a sine wave as you step through the order. If the peak of the cycle
occurs at the fi rst case and repeats every fi ve you will only collect the high
values, and the mean will be much larger than the population mean. Similarly,
if the trough occurs in the fi rst sample and the cycle length is fi ve, you collect
only the lowest values from the population, and the sample mean will be
too low.
(e) Cluster sampling is another way that may be more convenient than simple
random sampling. For example, when the Census Bureau does survey sam-
pling in a city, it may be convenient to sample every house on a particular
block since the blocks form a list that can be randomly sampled. In such situ-
ations, cluster sampling has advantages.
(f) Bootstrap sampling is not a procedure to sample from the population per
se. Instead, we have a sample (presumably random), and bootstrapping is
sampling with replacement from this sample to try to infer properties of
the population based on the variability of the bootstrap samples. In the
ordinary case when the sample size is n , we also take n elements for the boot-
strap sample by sampling with replacement from the n elements in the original
sample.- How does bootstrap sampling differ from simple random sampling?
As described earlier, bootstrap samples with replacement from a random sample,
whereas simple random sampling samples without replacement from a
population. - What is rejection sampling and how is it used?
Rejection sampling is a method for sampling at random without replacement. A
common way to sample without replacement is to eliminate the elements from the
population as they are selected in sequence and randomly sample each time from
the reduced population. With rejection sampling you can achieve the same proper-
ties without changing the population you draw the samples from. You simply keep
a running list of all the elements that have thus far been sampled, and if the new
one is a repeat of one of the old ones, you throw it out and try again always making
sure that nothing repeats. - Why is the sampling design choice more critical than the size of the sample?
If you make a bad choice of design you can create a large bias that cannot
be overcome by an increase in sample size no matter how large you make it.
However if the sample size is too small but the design is appropriate, you
can obtain unbiased estimates of the population parameters. Increasing the sample
size will not prevent us from obtaining an unbiased estimate, and since the
accuracy of an unbiased estimate depends only on its variance, the sample
size increase will reduce the variance and make the estimate more accurate.
So with a good design, we can improve the estimate by increasing the sample
size. But no increase in sample size will remove a bias that is due to the poor
design.
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