Statistical Methods for Data Science Mini Project 5

Statistical Methods for Data Science
Mini Project 5
Instructions:
• Total points = 20.
• Submit a typed report.

• Do a good job.
• You must use the following template for your report:
Mini Project #
Name
Names of group members (if applicable)
Contribution of each group member
Section 1. Answers to the specific questions asked
Section 2: R code. Your code must be annotated. No points may be given if a brief
look at the code does not tell us what it is doing.
1. Consider the data stored in bodytemp-heartrate.csv on eLearning, containing measurements of body temperature and heart rate for 65 male (gender = 1) and 65
female (gender = 2) subjects.
(a) Do males and females differ in mean body temperature? Answer this question
by performing an appropriate analysis of the data, including an exploratory
analysis.
(b) Do males and females differ in mean heart rate? Answer this question by performing an appropriate analysis of the data, including an exploratory analysis.
(c) Is there a linear relationship between body temperature and heart rate? Does
this relationship depend on gender? Answer these questions by performing an
appropriate analysis of the data, including an exploratory analysis.
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2. The goal of this exercise to see how large n should be for the large-sample and the
(parametric) bootstrap percentile method confidence intervals for the mean of an
exponential population to be accurate. To be specific, let X1, . . . , Xn represent a
random sample from an exponential (λ) distribution. Note that this distribution is
skewed and its mean is µ = 1/λ. We can construct two confidence intervals for µ —
one the large-sample z-interval (interval 1) and the other a (parametric) bootstrap
percentile method interval (interval 2). We would like to investigate their accuracy,
i.e., how close their estimated coverage probabilities are to the assumed nominal
level of confidence, for various combinations of (n, λ). This investigation will focus
on 1 − α = 0.95, λ = 0.01, 0.1, 1, 10 and n = 5, 10, 30, 100. Thus, we have a total of
4 ∗ 4 = 16 combinations of (n, λ) to investigate.
(a) For a given setting, compute Monte Carlo estimates of coverage probabilities of
the two intervals by simulating appropriate data, using them to construct the
two confidence intervals, and repeating the process 5000 times.
(b) Repeat (a) for the remaining combinations of (n, λ). Present an appropriate
summary of the results.
(c) Interpret all the results. Be sure to answer the following questions: In case of
the large-sample interval, how large n is needed for the interval to be accurate?
Likewise, in case of the bootstrap interval, how large n is needed for the interval
to be accurate? Do these answers depend on λ? Can we say that one method is
more accurate than the other? Which interval would you recommend? Provide
justification for all your conclusions.
(d) Do your conclusions in (c) depend on the specific values of λ that were fixed in
advance? Explain.
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