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Statistics Applied and Conceptual Statistics Worksheet
Need Questions Below Answered: Please also use textbook https://hastie.su.domains/ISLR2/ISLRv2_website.pdf for more clear ask.
Chapter 4
Questions 5,6,7,8,9,12,(13,14,15 – required R code)
We now examine the differences between LDA and QDA.
(a) If the Bayes decision boundary is linear, do we expect LDA or QDA to perform better on the training set? On the test set?
(b) If the Bayes decision boundary is nonlinear, do we expect LDA or QDA to perform better on the training set? On the test set?
(c) In general, as the sample size n increases, do we expect the test prediction accuracy of QDA relative to LDA to improve, decline, or be unchanged? Why?
(d) True or False: Even if the Bayes decision boundary for a given problem is linear, we will probably achieve a superior test er ror rate using QDA rather than LDA because QDA is flexible enough to model a linear decision boundary. Justify your answer.
Suppose we collect data for a group of students in a statistics classwith variables X1 = hours studied, X2 = undergrad GPA, and Y =receive an A. We fit a logistic regression and produce estimated ˆˆˆhas an undergrad GPA of 3.5 gets an A in the class.
(b) How many hours would the student in part (a) need to study to
have a 50 % chance of getting an A in the class?
̄
while the mean for those that didn’t was X = 0. In addition, the variance of X for these two sets of companies was σˆ2 = 36. Finally, 80 % of companies issued dividends. Assuming that X follows a nor mal distribution, predict the probability that a company will issue a dividend this year given that its percentage profit was X = 4 last year.
Hint: Recall that the density function for a normal random variable
coefficient, β0 = −6, β1 = 0.05, β2 = 1.
(a) Estimate the probability that a student who studies for 40 h and
mean value of X for companies that issued a dividend was X = 10, ̄
1 −(x−μ)2 /2σ2 is f(x) = √2πσ2 e
. You will need to use Bayes’ theorem.
4.8 Exercises 191
192 4. Classification
First we use logistic regression and get an error rate of 20 % on the training data and 30 % on the test data. Next we use 1nearest neigh bors (i.e. K = 1) and get an average error rate (averaged over both test and training data sets) of 18%. Based on these results, which method should we prefer to use for classification of new observations? Why?
(a) On average, what fraction of people with an odds of 0.37 ofdefaulting on their credit card payment will in fact default?
(b) Suppose that an individual has a 16% chance of defaulting on her credit card payment. What are the odds that she will de fault?
(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).
(e) Repeat (d) using LDA.
(f) Repeat (d) using QDA.
(g) Repeat (d) using KNN with K = 1.
(h) Repeat (d) using naive Bayes.
(i) Which of these methods appears to provide the best results on this data?
(j) Experiment with different combinations of predictors, includ ing possible transformations and interactions, for each of the methods. Report the variables, method, and associated confu sion matrix that appears to provide the best results on the held out data. Note that you should also experiment with values for K in the KNN classifier.
4.8 Exercises 193
194 4. Classification
(a) Create a binary variable, mpg01, that contains a 1 if mpg contains a value above its median, and a 0 if mpg contains a value below its median. You can compute the median using the median() function. Note you may find it helpful to use the data.frame() function to create a single data set containing both mpg01 and the other Auto variables.
(b) Explore the data graphically in order to investigate the associ ation between mpg01 and the other features. Which of the other features seem most likely to be useful in predicting mpg01? Scat terplots and boxplots may be useful tools to answer this ques tion. Describe your findings.
(c) Split the data into a training set and a test set.
(d) Perform LDA on the training data in order to predict mpg01 using the variables that seemed most associated with mpg01 in (b). What is the test error of the model obtained?
(e) Perform QDA on the training data in order to predict mpg01 using the variables that seemed most associated with mpg01 in (b). What is the test error of the model obtained?
(f) Perform logistic regression on the training data in order to pre dict mpg01 using the variables that seemed most associated with mpg01 in (b). What is the test error of the model obtained?
(g) Perform naive Bayes on the training data in order to predict mpg01 using the variables that seemed most associated with mpg01 in (b). What is the test error of the model obtained?
(h) Perform KNN on the training data, with several values of K, in order to predict mpg01. Use only the variables that seemed most associated with mpg01 in (b). What test errors do you obtain? Which value of K seems to perform the best on this data set?
(a) Write a function, Power(), that prints out the result of raising 2 to the 3rd power. In other words, your function should compute 23 and print out the results.
Hint: Recall that x^a raises x to the power a. Use the print() function to output the result.
(b) Create a new function, Power2(), that allows you to pass any two numbers, x and a, and prints out the value of x^a. You can do this by beginning your function with the line> Power2 < function(x, a) {
You should be able to call your function by entering, for instance,
> Power2(3, 8)
on the command line. This should output the value of 38, namely, 6, 561.
(c) Using the Power2() function that you just wrote, compute 103, 17 3
8 ,and131.
(d) Now create a new function, Power3(), that actually returns the result x^a as an R object, rather than simply printing it to the screen. That is, if you store the value x^a in an object called result within your function, then you can simply return() this result, using the following line:return(result)
The line above should be the last line in your function, beforethe } symbol.
(e) Now using the Power3() function, create a plot of f(x) = x .
return()
The xaxis should display a range of integers from 1 to 10, and 2
the yaxis should display x . Label the axes appropriately, and use an appropriate title for the figure. Consider displaying either the xaxis, the yaxis, or both on the logscale. You can do this by using log = “x”, log = “y”, or log = “xy” as arguments to the plot() function.
(f) Create a function, PlotPower(), that allows you to create a plot of x against x^a for a fixed a and for a range of values of x. For instance, if you call
> PlotPower(1:10, 3)
4.8 Exercises 195
2
then a plot should be created with an xaxis taking on values 333
1,2,…,10, and a yaxis taking on values 1 ,2 ,…,10 .
Statistics Applied and Conceptual Statistics Worksheet
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Statistics Applied and Conceptual Statistics Worksheet 
Statistics Applied and Conceptual Statistics Worksheet