STA302/1001代做、代写R设计、代做data、代做R编程语言

日期：2020-04-03 11:02

STA302/1001 - Assignment # 4

Due Friday April 10 by 11:59PM on Crowdmark

Student 1 Name:

Student 1 Email:

Student 2 Name:

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Instructions:

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you can both submit the assignment or edit submissions. Only one assignment should be submitted

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The assignment is divided into four questions, each with subparts. Each question needs to be uploaded

under the correct section in Crowdmark, otherwise it may be overlooked when graded. One

question is a calculation-type question, one will be a theoretical/derivation/proof type question, and

one will involve using R. You should make sure to show all your work with the first two questions,

while the R questions should be presented in a report-type format (i.e. include output and graphs

with written explanations of answers in the main document, R code places in an appendix at the

end). If you are comfortable with RMarkdown, it is recommended to complete your assignment

with it. Otherwise, any word processing document will suffice for the R question. You may submit

handwritten answers for questions 1 and 2, but they must be legible and neat.

Note that there is a 20% per day late penalty on assignments. After 48 hours of being late, the

assignment will no longer be accepted. This means that you should submit your assignment no

later than Sunday April 12 at 11:59PM to avoid receiving a grade of zero.

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Question 1 (14 points) - Derivations/Proof Question

This question walks you through how to prove the equivalence of two of the formulae for the

Cook’s Distance using matrix notation. Suppose X(i)

is the (n ? 1) × (p + 1) design matrix with

observation i removed, and X is the n × (p + 1) design matrix with all observations. Let xi be

a column vector representing the predictor values for observation i, and yi

is the response value

(scalar) for observation i. Finally, β? is the column vector of predictors estimated using the full data,

and β?

(i)

is the column vector of predictors estimated without using observation i.

(a) (2 points) Show how we would predict the response for observation i using the regression

model that has been fit without observation i.

(b) (3 points) Using the fact that β? = (X0X)

?1X0Y, and the following result,

show that post-multiplication of this result by (X0Y ? xiyi) yields

β?(i) = β? ? (X0X)?1xiyi +(X0X)?1xix.

(d) (4 points) Using your result in (c),

Question 2 (16 points) - Hand Calculation Question

We previously considered building multiple linear regression models for gas mileage of cars based

on characteristics of each vehicle model. We can now consider a few different models and attempt

to determine which model is better.

(a) (4 points) Using the table of summary values below, and that we have taken a sample of 30

vehicles, compute the AIC for each of the three models. Based on these values, which model

would you say is better?

Model Predictors Residual Standard Error

Model 1 all 11 predictors 3.227

Model 2 Displacement, Horsepower, Torque, Number of

Transmission Speeds, Weight 3.245

Model 3 Displacement, Horsepower, Weight 3.171

(b) (4 points) Using the above summary table, calculate the corrected AIC for each of the above

models. Based on this, would we prefer the same model as in part (a)?

(c) (4 points) Now, knowing that the sample variance of gas mileage is 39.28 MPG, find the

adjusted coefficient of determination for each of the models in (a). Based on this measure,

which model is preferred?

(d) (4 points) Suppose we consider the smallest model (model 3 from part (a)). We can fit a

model using each predictor as a response using the remaining predictors as predictors. Below

is a summary of each of these models.

Response Predictors Residual SE Sample Variance of Response

Displacement Horsepower, Weight 27.21 13511.05

Horsepower Displacement, Weight 15.64 1993.689

Weight Displacement, Horsepower 299.1 885420.2

Find the Variance Inflation Factor of each predictor. Should we be concerned about multicollinearity

in the model 3 from (a)?

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Question 3 (14 points) - R Data Analysis Question

Use the dataset found on Quercus under Assignment 4 materials to answer the below questions.

These data contain information on 50 men, on which were measured their percent body fat, their

height, their waist size, and their chest size.

(a) (1 points) Fit a multiple linear model to predict Percent Body Fat (Pct.BF) from waist size,

height and chest size.

(b) (3 points) Determine whether there are any:

(a) leverage points

(b) outlier observations

(c) influential observations. If there are, in what way do they influence the regression surface?

(c) (3 points) Can we use our residual plots to make conclusions regarding which assumptions

are violated? Support your answer using appropriate plots.

(d) (3 points) Determine whether the model from (a) is a valid model using appropriate plots.

(e) (2 points) Determine whether there is multicollinearity among the predictor variables.

(f) (2 points) Create an indicator variable for each of the observations 19 and 38. Add these

indicator variables to the model in (a) as main effect terms. Show that we have now removed

the influence of these observations from the model by showing that they no longer appear as

influential observations.

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