MA308: Statistical Calculation and Software
Assignment 2 (Oct 9– Nov 6, 2019)
2.1 For the “galton” dataset from Using R package,
(a) What will be the conclusion for testing the height of the child at ↵ = 0.05 level
of significance,
H0 : µ = 68, v.s. H1 : µ 6= 68,
given that variance is known to be 1.7873.
(b) If the variance is unknown in (a), carry out the likelihood-ratio test and draw
the conclusion at ↵ = 0.05 level of significance. Compare the result with that
of using t-test.
(c) Test whether the height of children and parents have the same mean value at
↵ = 0.05 level of significance. What if there is a “pairing” between the height
of the child and parent?
(d) In order to understand how parent’s height e↵ect a child’s height, first obtain a
scatter plot for child against parent, then obtain the Nadaraya-Watson kernel
estimator with the choice of two di↵erent kernels by implementing NadarayaWatson
Kernel Regression analysis.
(e) Test whether the spread of heights for the “parent” group and “child” group
are the same or not.
2.2 This question should be answered using the “Carseats” data set.
(a) Test whether Sales follow normal distribution.
(b) Fit a multiple regression model to predict Sales using Price, Urban, and US.
(c) Provide an interpretation of each coecient
in the model. Be careful some of
the variables in the model are qualitative!
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(d) Write out the model in equation form, being careful to handle the qualitative
variables properly.
(e) For which of the predictors can you reject the null hypothesis H0 : j
= 0?
(f) On the basis of your response to the previous question, fit a smaller model that
only uses the predictors for which there is evidence of association with the
outcome.
(g) How well do the models in (b) and (f) fit the data?
(h) Using the model from (f), obtain 95% confidence intervals for the coecient(s).
(i) Is there evidence of outliers or high leverage observations in the model from (f)?
(j) There is an indicator “Urban” in the “Carseat” data set, compare the mean
Sales of the “Urban” area with that of the “Rural” area, show the results of
the likelihood ratio test and the Mann-Whitney test for testing the equality of
these two mean values. Can we use the Wilcoxon’s Signed-Rank test? Why?
2.3 This question should be answered using the weekly.csv data set.
(a) Produce some numerical and graphical summaries of the Weekly data. Do there
appear to be any patterns?
(b) Use the full data set to perform a logistic regression with Direction as the
response and the five lag variables plus Volume as predictors. Use the summary
function to print the results. Do any of the predictors appear to be statistically
significant? If so, which ones?
(c) Compute the confusion matrix and overall fraction of correct predictions. Explain
what the confusion matrix is telling you about the types of mistakes made
by logistic regression.
(d) Now fit the logistic regression model using a training data period from 1990 to
2008, with Lag2 as the only predictor. Compute the confusion matrix and the
overall fraction of correct predictions for the held out data (that is, the data
from 2009 and 2010).
2.4 The “galaxies” data set from MASS package the velocities of 82 galaxies from six
well-separated conic sections of space (Postman et al., 1986, Roeder, 1990). The
data are intended to shed light on whether or not the observable universe contains
superclusters of galaxies surrounded by large voids. The evidence for the existence of
superclusters would be the multimodality of the distribution of velocities. Construct
a histogram of the data and add a variety of kernel estimates of the density function.
Estimate the density function of the “galaxies” data using histogram smoothing,
and uniform, Epanechnikov, biweight, and Gaussian kernels. What do you conclude
about the possible existence of superclusters of galaxies?