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Sample Paper Related To R Programming

Exercise 1.

Load the ncaa2018.csv data set and create histograms, QQ-norm and box-whisker plots for ELO. Add a title

to each plot, identifying the data.


Exercise 2.

Review Exercise 1, Homework 6, where you calculated skewness and kurtosis. The reference for this exercise,

https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm,


The following example shows histograms for 10,000 random numbers generated from a normal, a double exponential, a Cauchy, and a Weibull distribution.


We will reproduce the histograms for these samples, and add qqnorm and box-whisker plots.


Part a

Use the code below from lecture to draw 10000 samples from the normal distribution.

norm.sample <- rnorm(10000, mean=0, sd=1)

Look up the corresponding r* functions in R for the Cauchy distribution (use location=0, scale=1), and the Weibull distribution (use shape = 1.5). For the double exponential, use you can use the *laplace functions from the rmutil library, or you can use rexp(10000) - rexp(10000)


Draw 10000 samples from each of these distributions. Calculate skewness and kurtosis for each sample. You may use your own function, or use the moments library.


Part b

Plot the histograms for each distribution. Use par(mfrow=c(2,2)) in your code chunk to combine the four histogram in a single plot. Add titles to the histograms indicating the distribution. Set the x-axis label to show the calculated skewness and kurtosis, i.e. skewness = ####, kurtosis = ####

par(mfrow=c(2,2))

Part c

Repeat Part b, but with QQ-norm plots.

par(mfrow=c(2,2))

Part d

Repeat Part b, but with box-whisker plots.

par(mfrow=c(2,2))


Hints for SAS. If you create the samples in IML, use

Normal = j(1, 10000, .);

call randgen(Normal, "NORMAL", 0, `);


You can generate samples in the data step using

do i = 1 to 10000;

Normal = rand('NORMAL',0,1);

output;

end;


RAND doesn’t provide a Laplace option, but you can create samples from this distribution by

rand('EXPONENTIAL')-rand('EXPONENTIAL');

To group multiple plots, use

ods graphics / width=8cm height=8cm;

ods layout gridded columns=2;

ods region;

... first plot


ods region;

... second plot


ods layout end;


You might need to include


ods graphics off;

ods graphics on;

ODS GRAPHICS / reset=All;


to return the SAS graphics output to normal.



Exercise 3.

We will create a series of graphs illustrating how the Poisson distribution approaches the normal distribution with large λ. We will iterate over a sequence of lambda, from 2 to 64, doubling lambda each time. For each lambda draw 1000 samples from the Poisson distribution.


Calculate the skewness of each set of samples, and produce histograms, QQ-norm and box-whisker plots. You can use par(mfrow=c(1,3)) to display all three for one lambda in one line. Add lambda=## to the title of the histogram, and skewness=## to the title of the box-whisker plot.


Part b.

Remember that lambda represents the mean of a discrete (counting) variable. At what size mean is Poisson data no longer skewed, relative to normally distributed data? You might run this 2 or 3 times, with different seeds; this number varies in my experience.


par(mfrow=c(1,3))

If you do this in SAS, create a data table with data columns each representing a different μ. You can see combined histogram, box-whisker and QQ-norm, for all columns, by calling


proc univariate data=Distributions plot;
run;

At what μ is skewness of the Poisson distribution small enough to be considered normal?


Exercise 4

Part a

Write a function that accepts a vector vec, a vector of integers, a main axis label and an x axis label. This function should 1. iterate over each element i in the vector of integers 2. produce a histogram for vec setting the number of bins in the histogram to i 3. label main and x-axis with the specified parameters. 4. label the y-axis to read Frequency, bins = and the number of bins.

Hint: You can simplify this function by using the parameter ... - see ?plot or ?hist


Part b

Test your function with the hidalgo data set (see below), using bin numbers 12, 36, and 60. You should be able to call your function with something like


plot.histograms(hidalgo.dat[,1],c(12,36,60), main="1872 Hidalgo issue",xlab= "Thickness (mm)")

to plot three different histograms of the hidalgo data set.

If you do this in SAS, write a macro that accepts a table name, a column name, a list of integers, a main axis label and an x axis label. This macro should scan over each element in the list of integers and produce a histogram for each integer value, setting the bin count to the element in the input list, and labeling main and x-axis with the specified parameters. You should label the y-axis to read Frequency, bins = and the number of bins.


Test your macro with the hidalgo data set (see below), using bin numbers 12, 36, and 60. You should be able to call your macro with something like


%plot_histograms(hidalgo, y, 12 36 60, main="1872 Hidalgo issue", xlabel="Thickness (mm)");

to plot three different histograms of the hidalgo data set.

Hint: Assume 12 36 60 resolve to a single macro parameter and use %scan. Your macro definition can look something like


%macro plot_histograms(table_name, column_name, number_of_bins, main="Main", xlabel="X Label")

Data

The hidalgo data set is in the file hidalgo.dat These data consist of paper thickness measurements of stamps from the 1872 Hidalgo issue of Mexico. This data set is commonly used to illustrate methods of determining the number of components in a mixture (in this case, different batches of paper). See


https://www.jstor.org/stable/2290118,

https://books.google.com/books?id=1CuznRORa3EC&lpg=PA95&pg=PA94#v=onepage&q&f=false and

https://books.google.com/books?id=c2_fAox0DQoC&pg=PA180&lpg=PA180&f=false .


Some analysis suggest there are three different mixtures of paper used to produce the 1872 Hidalgo issue; other analysis suggest seven. Why do you think there might be disagreement about the number of mixtures?


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