Expert answer:UMUC STATS 200, FALL 2017 QUIZ 1
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UMUC STATS 200, FALL 2017 QUIZ 1
KOBI SNITZ
(1) Suppose you want to design a poll to predict the outcome of the presidential
elections. Explain how you would sample the population for the poll. Say
what type of sample would you use and explain specifically how you would
design such a sample. Explain why such a sample is the appropriate one
for this situation.
(2) After describing your honest poll explain how a dishonest pollster can design
a poll that might look fair but is likely to produce a result that they prefer.
(3) Consider a monthly sample of the water temperature in the Pacific ocean
in a calendar year measured in celsius degrees.
{7.2, 6.7, 7.3, 7.3, 7.7, 8.2, 10.1, 11.3, 11.2, 10.0, 8.8, 7.5}
What is the variance and standard deviation of this sample. Based on
your calculation would you consider a measurement of 5 degrees unusual ?
would a measurement of 11.5 degrees be more or less likely in your opinion?
Explain your reasoning.
(4) Consider the list of annual rates of growth of an olive tree expressed in
percentiles.
{5, 4.5, 4.3, 5.6, 3.6}
What would be the size of an olive tree which was 1 meter tall after 5 years
of these growth rates ? (hint: after the first year the height of the tree
would be 1.05 meters.)
Express the growth rates in multiplicative factors. That is, the first year
the height of the tree is multiplied by 1.05 because it grows by 5%. Take
the geometric mean of the multiplicative growth rates. Why is that mean
appropriate for this situation ?
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UMUC STATS 200, FALL 2016 QUIZ 1 SOLUTION
KOBI SNITZ
(1) Suppose you want to design a poll to predict the outcome of the presidential
elections. Explain how you would sample the population for the poll. Say
what type of sample would you use and explain specifically how you would
design such a sample. Explain why such a sample is the appropriate one
for this situation.
Solution: In order to predict the outcome of the US presidential elections we need a stratified sample of the population. The population should
be stratified in several ways. First of all because of the electoral college
system, the sample should be stratified by states. Secondly there are many
factors which might cause a sample to be biased and you would want to
eliminate bias from your sample by stratifying the sample. Such factors
can be: likely voters, age, income, education level and many more factors.
Typically political polling is so complex that choosing the right sample is an
art practiced by specialists who continuously calibrate their sampling method
(2) After describing your honest poll explain how a dishonest pollster can design
a poll that might look fair but is likely to produce a result that they prefer.
Solution: To take just one example of many, suppose a pollster wanted
to please their client by giving them good news about their chances of being
elected. The pollster knows that their candidate client is more popular with
older and more affluent voters. To create a sympathetic sample the pollster
might randomly select people from a list of home owners who tend to be
more affluent and older
(3) Consider a monthly sample of the water temperature in the Pacific ocean
in a calendar year measured in celsius degrees.
{7.2, 6.7, 7.3, 7.3, 7.7, 8.2, 10.1, 11.3, 11.2, 10.0, 8.8, 7.5}
What is the mean and standard deviation of this sample. Based on your calculation would you consider a measurement of 5 degrees unusual ? would a
measurement of 11.5 degrees be more or less likely in your opinion? Explain
your reasoning.
Solution: The mean of this sample is x̄ = 8.608 and the sample standard
deviation is s = 1.636. Using these two values we can see that a measurement of 5 degrees is more than 2 standard deviations away from the mean
which is unusual. We can also express this by calculating the z-score of 5
which is z5 = 5−x̄
s = −2.206. The z score allows us to compare how ’unusual’ two measurements are. To compare a measurement of 5 degrees to
11 degrees we calculate the z-score of 11. z11 = 11−x̄
= 1.462. Since the
s
z-score of 11 is smaller in magnitude than the z-score of 5, a measurement
of 11 degrees is less far from the mean and therefore less unusual
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KOBI SNITZ
(4) Consider the list of annual rates of growth of an olive tree expressed in
percentiles.
{5, 4.5, 4.3, 5.6, 3.6}
What would be the size of an olive tree which was 1 meter tall after 5 years
of these growth rates ? (hint: after the first year the height of the tree
would be 1.05 meters.)
Express the growth rates in multiplicative factors. That is, the first year
the height of the tree is multiplied by 1.05 because it grows by 5%. Take
the geometric mean of the multiplicative growth rates. Why is that mean
appropriate for this situation ?
Solution: The height of the tree after 5 years is the product of the growth
rate at every year. To see that, consider that at the second year the tree
grows by %4.5 which of its size at the beginning of that year. In other
words the size of the tree is multiplied by 1.045. That size in turn is the
initial size of the tree multiplied by the grows factor of the first year (1.05).
In total after the first two years the tree grows by 1.045 ∗ 1.05 = 1.09725
and so on for the five years. The total growth of the tree after 5 years is
1.05 ∗ 1.045 ∗ 1.043 ∗ 1.056 ∗ 1.036 = 1.252
The geometric mean of these numbers is 1.046. The geometric mean has
the property that if the growth factor at every year was it, the grows over
5 years will be the same as the combination of the individual growth rates.
Note that the algebraic mean (which you might know as the usual average)
does not have that property of the geometric mean.
…
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