OLLSCOIL NA hEIREANN, CORCAIGH
THE NATIONAL UNIVERSITY OF IRELAND, CORK
COLAISTE NA hOLLSCOILE, CORCAIGH
UNIVERSITY COLLEGE, CORK
ST4060 – ST6015 – ST6040
Continuous assessment 1 – 2020-21
Eric Wolsztynski
eric.
List of (possibly) useful R functions:
apply()
boxplot()
c()
coef()
fitted()
lm()
matrix()
mean()
nrow()
par()
quantile()
rgamma()
round()
sample()
set.seed()
summary()
unique()
Question 1
Please run set.seed(6040) before running the analysis below.
Implement a Monte Carlo simulation with M = 1, 000 repetitions to demonstrate that the number of data
points selected in any bootstrap resample is about 63.2%. Consider a sample size N = 1, 000 for this Monte
Carlo experiment.
Note: no dataset is required to perform this analysis. Hint: R function unique() may be useful here.
Question 2
Please run set.seed(6015) before running the analysis below.
Implement a Monte Carlo simulation of M = 1, 000 random samples of N = 100 realizations of the Gamma
distribution G(a, b) with shape a = 3 and rate b = 2. Calculate and store the M corresponding sample
means.
(a) Quote the Monte Carlo estimate of the expected value of the sample mean for this probability model.
(b) Is this value of the sample mean surprising? Briefly explain why (or why not).
(c) Quote the Monte Carlo estimate of the standard error of the sample mean for this probability model.
Question 3
Please run set.seed(4060) before running the analysis below.
Consider R’s dataset trees of 31 felled black cherry trees. Generate 100 bootstrap estimates of the slope
estimate in the linear regression of tree height (Height, in feet) with respect to tree diameter (Girth, in
inches), i.e. to explain tree height in function of tree girth.
(a) Provide a boxplot of the sampling distribution of bootstrap estimates of the parameter of interest.
(b) Quote the bootstrap estimate of the expected value of the least squares estimator of the regression
slope parameter.
(c) Quote the bootstrap estimate for the standard error associated with the least squares estimator of the
regression slope parameter.
(d) Provide an empirical bootstrap confidence interval for the true value of the regression slope parameter.
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