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The
Analysis of Two Related Variables
The goal of this
assignment is to analyze two
quantitative variables (that may or may not be related to each other)
to see if
you can predict one from the other. The data can either be from a
census, a
survey, or an experiment. It may be real data you found in a published
source
(not the text book, and not one that has already analyzed the data!) or
it may
be data you gathered yourself. It may be data related to a master’s or
senior
honor’s thesis, but it may not be data from a previous statistics
course. Both
variables need to be continuous (or at least have a large number of
different
levels if discrete).
The goal is to analyze
the data and to present the
results so that someone who had not had a statistics course could
understand
them. For example, if you use the median as a descriptive statistic you
would
need to briefly explain to the reader what the median is and why you
chose it.
When you report the p-value of a hypothesis test you need to explain
what it
means, and why you would probably accept or reject the null hypothesis.
You
don't need to explain how the tests of hypotheses work, but you do need
to
explain what the assumptions are.
The
project will have
to address five main questions:
1) What question are
you trying to answer? (e.g. Can
the height of students be used to predict how far
someone can jump.)
2) Why is this
question
of interest? (e.g. In
grade school one of the tests in gym class is to see how far
you can
jump. Is
this fair to people who are short?)
3) How
was the data
gathered, and what limitations
does this imply? How would you overcome these
limitations. (e.g. Only
students in the fifth period gym class were used, this is bad because…)
4) Describe the two
variables individually. (e.g. The
average height was… Jumping distance was
skewed right….)
5) Describe the
relationship between the variables.
(e.g. The jumping distance is estimated to increase by …for each
additional
inch of height…)
The paper should be
typed, using complete sentences, and transition between the
various sections. If you are using
data collected by someone else, reference the source appropriately. The
paper
should be between 3 and 5 pages long, excluding any graphs.
The project is due 4:00 pm on Wednesday,
December 6th. You
must get approval of
the
data you are using by Monday, November 20th.
You should plan on
having a
friend read over both the assignment and project before handing them in
to make
sure you have answered the questions.
In the past, students
have chosen inappropriate data
(not continuous for example) or done the analysis in reverse (predicted
x from y instead of y from x).
Both of these
are grounds for
failing the project outright. Forgetting to answer several of the
questions
also results in low grades.
Specifics:
3)
If the data comes
from a sample: Define the
desired target population and describe how the sample was collected. If
you
were not able to sample from the desired population, state what
differences you
might expect between the population that was actually sampled from and
the
desired target population. If you were not able to take a simple random
sample
(page 150) from the population, discuss how the sampling could be
improved if
you were allowed more money and time.
If the data comes
from an experiment: Describe
how the experiment was carried out, describe any sources of extra
variation (e.g.
changing temperature, different people making the measurements,
etc...). Did
you try to control these? Discuss how the experiment could be improved
if you
were allowed (more) money and time.
If the data is
census data or fixed measurement
(e.g. area of states or reported death rates): Describe how the
measurements were gathered, how these measurements have changed over
time, what
some alternate census levels are (e.g. State vs. PMSA), whether it
seems to be
reasonable that these results will be the same in the future, and
whether the
variables need to be adjusted to remove the effects of other variables
such as
total population or cost of living.
4)
When describing
the variables individually, give
the appropriate plots and descriptive statistics to succinctly, but
thoroughly
describe the data. That is, decide which of the graphs and statistics
best describe the data to the reader. Give the reader help in
interpreting the
graphs and statistics by telling them what they should be seeing.
If the data is from
an experiment or sample,
construct confidence intervals for the means of the variables.
Interpret them
and say if we can trust these intervals or not (that is, are the
assumptions
met).
5) The Model: Fit
a linear regression model to
your data. Be sure to state what model you are attempting to fit to the
data in
terms of the variables you are using.
Statistics: The
report of the regression you
performed should include the following statistics: the estimated
regression
line, a confidence interval for the slope, the p-value for testing that
the
slope is zero, the coefficient of determination (r2) , and the
standard error about the line
(square root of MSE). Make sure and tell the reader why these
statistics should
be useful to them, and interpret them in the context of your data set.
Graphics: Give
the scatter plot of the data
with the regression line.
Assumptions: Check
the assumptions needed for
the regression. If the assumptions are not met, then don’t forget to
point out
to the reader that they can’t entirely trust the confidence intervals
and
hypothesis tests you found when performing the regression. If you find
any
outliers, see if they have a significant effect on your results by
running the
regression again without them and seeing if your regression line
changes much.
Finally,
Don’t forget to include a short summary
at the end of
your paper to tie everything together!
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