# execrise question

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(chapter 2)

1. For each of parts (a) through (d), indicate whether we would generally

expect the performance of a flexible statistical learning method to be

better or worse than an inflexible method. Justify your answer.

(a) The sample size n is extremely large, and the number of predictors p is small.

(b) The number of predictors p is extremely large, and the number

of observations n is small.

(c) The relationship between the predictors and response is highly

non-linear.

(d) The variance of the error terms, i.e. ?2 = Var(), is extremely

high.

3. We now revisit the bias-variance decomposition.

(a) Provide a sketch of typical (squared) bias, variance, training error, test error, and Bayes (or irreducible) error curves, on a single plot, as we go from less flexible statistical learning methods

towards more flexible approaches. The x-axis should represent. the amount of flexibility in the method, and the y-axis should

represent the values for each curve. There should be five curves.

Make sure to label each one.

(b) Explain why each of the five curves has the shape displayed in

part (a).

6. Describe the differences between a parametric and a non-parametric

statistical learning approach. What are the advantages of a parametric approach to regression or classification (as opposed to a nonparametric approach)? What are its disadvantages?

7. The table below provides a training data set containing six observations, three predictors, and one qualitative response variable.

Obs. X1 X2 X3 Y

1 0 3 0 Red

2 2 0 0 Red

3 0 1 3 Red

4 0 1 2 Green

5 ?1 0 1 Green

6 1 1 1 Red

Suppose we wish to use this data set to make a prediction for Y when

X1 = X2 = X3 = 0 using K-nearest neighbors.

(a) Compute the Euclidean distance between each observation and

the test point, X1 = X2 = X3 = 0.

(b) What is our prediction with K = 1? Why?

(c) What is our prediction with K = 3? Why?

(d) If the Bayes decision boundary in this problem is highly nonlinear, then would we expect the best value for K to be large

(Chapter 3)

1. Describe the null hypotheses to which the p-values given in Table 3.4

correspond. Explain what conclusions you can draw based on these

p-values. Your explanation should be phrased in terms of sales, TV,

radio, and newspaper, rather than in terms of the coefficients of the

linear model.

3. Suppose we have a data set with five predictors, X1 = GPA, X2 = IQ,

X3 = Gender (1 for Female and 0 for Male), X4 = Interaction between

GPA and IQ, and X5 = Interaction between GPA and Gender. The

response is starting salary after graduation (in thousands of dollars).

Suppose we use least squares to fit the model, and get ?0 = 50, ?1 =

20, ?2 = 0.07, ?3 = 35, ?4 = 0.01, ?5 = ?10.

(a) Which answer is correct, and why?

i. For a fixed value of IQ and GPA, males earn more on average

than females.

ii. For a fixed value of IQ and GPA, females earn more on

average than males.

iii. For a fixed value of IQ and GPA, males earn more on average

than females provided that the GPA is high enough.

iv. For a fixed value of IQ and GPA, females earn more on

average than males provided that the GPA is high enough.

(b) Predict the salary of a female with IQ of 110 and a GPA of 4.0.

(c) True or false: Since the coefficient for the GPA/IQ interaction

term is very small, there is very little evidence of an interaction

effect. Justify your answer.

4. I collect a set of data (n = 100 observations) containing a single

predictor and a quantitative response. I then fit a linear regression

model to the data, as well as a separate cubic regression, i.e. Y =

?0 + ?1X + ?2X2 + ?3X3 + .

(a) Suppose that the true relationship between X and Y is linear,

i.e. Y = ?0 + ?1X + . Consider the training residual sum of

squares (RSS) for the linear regression, and also the training

RSS for the cubic regression. Would we expect one to be lower

than the other, would we expect them to be the same, or is there

not enough information to tell? Justify your answer.

(b) Answer (a) using test rather than training RSS.

(c) Suppose that the true relationship between X and Y is not linear,

but we dont know how far it is from linear. Consider the training

RSS for the linear regression, and also the training RSS for the

cubic regression. Would we expect one to be lower than the

other, would we expect them to be the same, or is there not

enough information to tell? Justify your answer.

(d) Answer (c) using test rather than training RSS.

7. It is claimed in the text that in the case of simple linear regression

of Y onto X, the R2 statistic (3.17) is equal to the square of the

correlation between X and Y (3.18). Prove that this is the case. For

simplicity, you may assume that ¯x = ¯y = 0.

(Chapter 4)

9. This problem has to do with odds.

(a) On average, what fraction of people with an odds of 0.37 of

defaulting 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 default?

(Chapter 5)

2. We will now derive the probability that a given observation is part

of a bootstrap sample. Suppose that we obtain a bootstrap sample

from a set of n observations.

(a) What is the probability that the first bootstrap observation is

not the jth observation from the original sample? Justify your

answer.

(b) What is the probability that the second bootstrap observation

is not the jth observation from the original sample?

(c) Argue that the probability that the jth observation is not in the

bootstrap sample is (1 ? 1/n)n.

(d) When n = 5, what is the probability that the jth observation is

in the bootstrap sample?

(e) When n = 100, what is the probability that the jth observation

is in the bootstrap sample?

(f) When n = 10, 000, what is the probability that the jth observation is in the bootstrap sample?

there will be other questions similair to these before the end of this practice to be answered in a timely manner.

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