[SOLVED] R

Gender Di erences in Wages (Revised based on ex 25 of Chapter 12 in The Statistical Sleuth). Display 12.21 is a partial listing of a data set with weekly earnings for 9,835 Americans surveyed in the March 2011 Current Population Survey (CPS). The dataset is stored in the object ex1225 of the R library Sleuth3. What evidence is there from these data that males tend to receive higher earnings than females with the same values of the other variables? Note that there might be an interaction between Sex and Marital Status. (Data from U.S. Bureau of Labor Statistics and U.S. Bureau of the Census: Current Population Survey, March 2011 http://www.bls.census.gov/cps ftp.html#cpsbasic; accessed July 25, 2011.)

Display taken from class text: The Statistical Sleuth.

In order to investigate the above problem, please use R to answer the following Questions 1 3 in the answer sheet.

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Question 1 (Multiple Linear Regression and Variable Selection, 2.0 points)

Consider the multiple linear regression model to regress the logarithm of Week- lyEarnings on variables Age, Sex, MaritalStatus and EdCode (please do not consider the interaction terms for now).

Please answer the following questions in the answer sheet.

a) (0.5 points) Please use R to obtain the fitted model based on the above variables. What is the least squares estimate for the coe cient of EdCode (rounded to four decimal places)? Please also interpret this estimated coe cient.

b) (0.5 points) Based on the summary function output of this fitted model, what are the null hypothesis and the alternative hypothesis for the F-statistic in the

summary function output? What conclusion can you obtain for this F -test?

c) (0.5 points) Please use R to perform the backward elimination based on F -statistic. Which variables should we choose to predict the logarithm of WeeklyEarnings by using this variable selection method?

d) (0.5 points) Please paste the R codes for all the above analyses of Question 1 in the answer sheet.

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Question 2 (Model Diagnostics, 3.5 points)
Consider the multiple linear regression model in Quesiton 1 a). Please answer the

following questions in the answer sheet.

1. a) (0.5 points) Based on the summary function output of the fitted model in Quesiton 1 a), please interpret the R-squared.
2. b) (0.5 points) Please paste the residuals versus fitted values plot of the fitted model in Quesiton 1 a) in the answer sheet. Are the assumptions in the multiple linear regression model violated based on this plot?
3. c) (0.5 points) Please paste the Q-Q plot of the residuals based on the fitted model in Quesiton 1 a) in the answer sheet. What conclusions can you obtain via the Q-Q plot?
4. d) (0.5 points) Please paste the Cooks distance plot of the fitted model in Quesiton 1 a) in the answer sheet. Based on the criterion introduced in lectures, are there any influential observations? Why or why not?
5. e) (0.5 points) Please find the observation with the largest Cooks distance. (Hint: use which function in R.) Based on the rule of thumb cut-o s for the studentized residual, is this observation an outlier? How to deal with this suspected influential observation?
6. f) (0.5 points) We have found the observation with the largest Cooks distance in e). Based on the rule of thumb cut-o for the leverage, does this observation have distant explanatory variable values? Why or why not?
7. g) (0.5 points) Please paste the R codes for all the above analyses of Question 2 in the answer sheet.

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Question 3 (Multiple Linear Regression for Continuous and Categorical Explanatory Variables, 3.0 points)

Consider the multiple linear regression model in Quesiton 1 a), but we would like to add more explanatory variables. Please answer the following questions in the answer sheet.

a) (0.5 points) We first use the following R codes to generate the indicator variables for categorical variables Region and MetropolitanStatus, respectively:

If we are also interested to show whether or not the mean of log(WeeklyEarnings) in each category of FedGov, StateGov and LocalGov, is significantly di erent from that in the category of Private, directly via the R output, which category should we choose as the baseline level for the categorical variable JobClass? Which indicator variables of JobClass should we select for model fitting to realise the above purpose?

b) (0.5 points) Please use R to obtain the fitted model based on all the variables involved in Question 1 a) and Question 3 a). Still please do not consider the interaction terms for now. Based on the summary function output of this fitted model, if we control the other variables, is the mean of log(WeeklyEarnings) in each category of FedGov, StateGov and LocalGov, is significantly di erent from that in the category of Private?

c) (0.5 points) Based on the fitted model in Question 3 b), now we are interested in testing whether or not at least one of categories of FedGov, StateGov and LocalGov has a di erent level of the mean of log(WeeklyEarnings), compared to the category of Private, when other variables are held constant. Please use R to obtain an appropriate test statistic and the corresponding p-value. What

conclusion can you obtain based on the result?

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d) (0.5 points) Consider the model and the variables in Question 3 b). But now we add an interaction between Sex and Marital Status and obtain a new model. Compute and show the sum of squared errors (SSE) for these two fitted models. Which one is smaller?

e) (0.5 points) Consider the model with the interaction in Question 3 d). What are the explanations of the estimated coe cient of the interaction term? Is the interaction between Sex and Marital Status significant? Why or why not?

f) (0.5 points) Please paste the R codes for all the above analyses of Question 3 in the answer sheet.

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Question 4 (Simulation for Multiple Linear Regression, 1.5 points)

Consider the multiple linear regression model {Y |X1, X2} = 0 + 1X1 + 2X2 for the observations {Yi, X1,i, X2,i}ni=1, and the least squares estimates 0, 1 and 2 for the coe cients 0, 1 and 2 can be obtained.

Lily wants to use R to generate random samples based on the multiple linear regression model assumptions. She follows the steps below.

Step 1: Specify 0 = 2, 1 = 1 and 2 = =1,
Step 2: Suppose the observations X1,1, , X1,n are 1, 2, , 100, so the number of

observations n = 100.
Step 3: Generate X2,1, , X2,n from the t3 distribution. (Hint: similar to the codes

on page 18 of Lecture Notes 3.)

Step 4: Generate E1, , En from the standard normal distribution [N(0,1) with mean 0 and variance 1].

Step 5: Generate Yi = {Yi|X1,i, X2,i} + Ei, i = 1, , n.
Step 6: Repeat Step 4 Step 5 1,000 times and obtain 1,000 di erent datasets of

{Yi, X1,i, X2,i}ni=1.

Lei Li is a friend of Lily. Lily hands over the above 1,000 datasets to him but she does not tell him the true values of 0, 1 and 3. Based on each dataset, Lei Li computes the least squares estimates 0, 1 and 2 as well as the 95% confidence interval for the mean of response given X1 = 2.5 and X2 = 0. Ultimately, he obtains 1,000 di erent confidence intervals.

Then Lily computes the mean of response {Y |X1 = 2.5, X2 = 0} and tells Lei Li this information. Lei Li counts the number of the confidence intervals that cover {Y|X1 =2.5,X2 =0}.

Please answer the following questions in the answer sheet.

a) (0.5 points) Suppose you play both roles of Lily and Lei Li and realise the above steps in R. Please paste the complete R codes for all the above procedures in the answer sheet. (Hint: similar to the codes on page 7 of Lecture Notes 2.)

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b) (0.5 points) What is the number of the confidence intervals that cover {Y |X1 = 2.5, X2 = 0} based on the above steps? Please answer this question in the answer sheet.

c) (0.5 points) Based on the result of b), interpret the 95% confidence interval for the mean of response. Please answer this question in the answer sheet.

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