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- 首页 >> 其他 STAT UN2102 Homework 4 [100 pts]
Due 11:59pm Monday, May 6th on Canvas
Your homework should be submitted on Canvas as an R Markdown file. Please submit
the knitted .pdf or .html file along with the .Rmd file. We will not (and cannot) accept
any other formats. Please clearly label the questions in your responses and support your
answers by textual explanations and the code you use to produce the result. We may print
out your homeworks. Please do not waste paper by printing the dataset or any vector over,
say, length 20.
Goals: Simulating probability distributions using the accept-reject method, simulating a
sampling distribution related to the linear regression model.
1 Reject-Accept Method
Let random variable X denote the temperature at which a certain chemical reaction takes
place. Suppose that X has probability density function
Perform the following tasks:
1. Determine the maximum of f(x). Find an envelope function e(x) by using a uniform
distribution for g(x) and setting e(x) = maxx{f(x)}.
2. Using the Accept-Reject Algorithm, write a program that simulates 1000 draws
from the probability density function f(x) from Equation 1.
3. Plot a histogram of your simulated data with the density function f overlayed in the
graph. Label your plot appropriately.
2 Regression and Empirical Size
2.1 Regression
We work with the grocery retailer dataset from Canvas. The description follows:
1A large national grocery retailer tracks productivity and costs of its facilities closely. Consider
a data set obtained from a single distribution center for a one-year period. Each data
point for each variable represents one week of activity. The variables included are number
of cases shipped in thousands (X1), the indirect costs of labor as a percentage of total
costs (X2), a qualitative predictor called holiday that is coded 1 if the week has a holiday
and 0 otherwise (X3), and total labor hours (Y ). Consider the multiple linear regression
model
(2) Yi = β0 + β1 Xi1 + β2 Xi2 + β3 Xi3 + i, i = 1, 2, . . . , 52,
and iid～ N(0, σ2).
Perform the following tasks:
4. Read in the grocery retailer dataset. Name the dataset grocery.
5. Use the least squares equation = (XTX)
1XTY to estimate regression model (2).
To estimate the model, use the linear model function in R, i.e., use lm().
6. Use R to estimate σ2, i.e., compute MSE =1
. To perform this task,
use the residuals function.
2.2 Test for Slope
Now consider investigating if the number of cases shipped (X1) is statistically related to
total labor hours (Y ). To investigate the research question, we run a t-test on the coefficient
corresponding to X1, i.e., we test the null alternative pair
(3) H0 : β1 = 0 versus HA : β1 6= 0.
To run the hypothesis testing procedure, we use the t-statistic
1 is the second element of the least squares estimator β= (XTX)
1XTY and
SE(β1) is the standard error of β?
1. The least squares estimates, estimated standard errors,
t-statistics and p-values for all coefficients β0, β1, β2, β3 are nicely organized in the standard
linear regression output displayed in Table 1. To get this output in R, use the summary()
function on your model.
Test the manager’s claim in (3) using the R functions lm() and summary().
2Table 1: Standard Multiple Linear Regression Output
Estimate Std. Error t value Pr(> |t|) or Sig
(Intercept) β
2.3 Sampling Distribution
Under model (2) and under the null hypothesis H0 : β1 = 0, the test statistic (4) has a
student’s t-distribution with n 4 degrees of freedom, i.e.,
The goal of this section is to simulate the sampling distribution of the t-statistic.
Perform the following tasks:
5. Write a loop that simulates the sampling distribution of the t-statistic under null
hypothesis (3) with the multiple linear regression model (2). To accomplish this task:
i. Assume the true model relating Y with X1, X2, X3 is
(5) Yi = 4200 + β1Xi1 ? 15X2 + 620X3 + i, i = 1, 2, . . . , 52,i
iid～ N(0, 20500).
ii. Assuming H0 : β1 = 0 is true, simulate 10,000 draws from model (5) using the
fixed covariates X2, X3.
iii. For each iteration of the loop, fit the full model
using the simulated Y and fixed covariates X1, X2, X3.
iv. For each iteration of the loop, also compute the t-statistic from equation (4).
Store these values in a vector t.stat. Hint: Use the summary function in R and
extract the actual summary table using the code summary(model)[[4]]. Then
extract the relevant t-statistic from the table.
v. Display the first six elements of your simulated t-values.
37. Plot a histogram of the simulated sampling distribution. Overlay the correct t-density
on this histogram, i.e., overlay the density t(df = 52 ? 4). Plot the density in green
and set breaks=40 in the histogram. Make sure to label the plot appropriately. You
can use base R or ggplot.
8. Recall that the significance level of a testing procedure is defined as
P(Type I error) = P(Rejecting H0 when H0 is true) = α.
The significance level is often called the size of the testing procedure. Based on
significance levels α = 0.10, 0.05, 0.01, compute the sample proportion of simulated
t-values that fell in the rejection region. The proportion of simulated rejected t-values
under the null is called the empirical size of a test. The three values should be close
to the actual α levels.
4
Due 11:59pm Monday, May 6th on Canvas
Your homework should be submitted on Canvas as an R Markdown file. Please submit
the knitted .pdf or .html file along with the .Rmd file. We will not (and cannot) accept
any other formats. Please clearly label the questions in your responses and support your
answers by textual explanations and the code you use to produce the result. We may print
out your homeworks. Please do not waste paper by printing the dataset or any vector over,
say, length 20.
Goals: Simulating probability distributions using the accept-reject method, simulating a
sampling distribution related to the linear regression model.
1 Reject-Accept Method
Let random variable X denote the temperature at which a certain chemical reaction takes
place. Suppose that X has probability density function
Perform the following tasks:
1. Determine the maximum of f(x). Find an envelope function e(x) by using a uniform
distribution for g(x) and setting e(x) = maxx{f(x)}.
2. Using the Accept-Reject Algorithm, write a program that simulates 1000 draws
from the probability density function f(x) from Equation 1.
3. Plot a histogram of your simulated data with the density function f overlayed in the
graph. Label your plot appropriately.
2 Regression and Empirical Size
2.1 Regression
We work with the grocery retailer dataset from Canvas. The description follows:
1A large national grocery retailer tracks productivity and costs of its facilities closely. Consider
a data set obtained from a single distribution center for a one-year period. Each data
point for each variable represents one week of activity. The variables included are number
of cases shipped in thousands (X1), the indirect costs of labor as a percentage of total
costs (X2), a qualitative predictor called holiday that is coded 1 if the week has a holiday
and 0 otherwise (X3), and total labor hours (Y ). Consider the multiple linear regression
model
(2) Yi = β0 + β1 Xi1 + β2 Xi2 + β3 Xi3 + i, i = 1, 2, . . . , 52,
and iid～ N(0, σ2).
Perform the following tasks:
4. Read in the grocery retailer dataset. Name the dataset grocery.
5. Use the least squares equation = (XTX)
1XTY to estimate regression model (2).
To estimate the model, use the linear model function in R, i.e., use lm().
6. Use R to estimate σ2, i.e., compute MSE =1
. To perform this task,
use the residuals function.
2.2 Test for Slope
Now consider investigating if the number of cases shipped (X1) is statistically related to
total labor hours (Y ). To investigate the research question, we run a t-test on the coefficient
corresponding to X1, i.e., we test the null alternative pair
(3) H0 : β1 = 0 versus HA : β1 6= 0.
To run the hypothesis testing procedure, we use the t-statistic
1 is the second element of the least squares estimator β= (XTX)
1XTY and
SE(β1) is the standard error of β?
1. The least squares estimates, estimated standard errors,
t-statistics and p-values for all coefficients β0, β1, β2, β3 are nicely organized in the standard
linear regression output displayed in Table 1. To get this output in R, use the summary()
function on your model.
Test the manager’s claim in (3) using the R functions lm() and summary().
2Table 1: Standard Multiple Linear Regression Output
Estimate Std. Error t value Pr(> |t|) or Sig
(Intercept) β
2.3 Sampling Distribution
Under model (2) and under the null hypothesis H0 : β1 = 0, the test statistic (4) has a
student’s t-distribution with n 4 degrees of freedom, i.e.,
The goal of this section is to simulate the sampling distribution of the t-statistic.
Perform the following tasks:
5. Write a loop that simulates the sampling distribution of the t-statistic under null
hypothesis (3) with the multiple linear regression model (2). To accomplish this task:
i. Assume the true model relating Y with X1, X2, X3 is
(5) Yi = 4200 + β1Xi1 ? 15X2 + 620X3 + i, i = 1, 2, . . . , 52,i
iid～ N(0, 20500).
ii. Assuming H0 : β1 = 0 is true, simulate 10,000 draws from model (5) using the
fixed covariates X2, X3.
iii. For each iteration of the loop, fit the full model
using the simulated Y and fixed covariates X1, X2, X3.
iv. For each iteration of the loop, also compute the t-statistic from equation (4).
Store these values in a vector t.stat. Hint: Use the summary function in R and
extract the actual summary table using the code summary(model)[[4]]. Then
extract the relevant t-statistic from the table.
v. Display the first six elements of your simulated t-values.
37. Plot a histogram of the simulated sampling distribution. Overlay the correct t-density
on this histogram, i.e., overlay the density t(df = 52 ? 4). Plot the density in green
and set breaks=40 in the histogram. Make sure to label the plot appropriately. You
can use base R or ggplot.
8. Recall that the significance level of a testing procedure is defined as
P(Type I error) = P(Rejecting H0 when H0 is true) = α.
The significance level is often called the size of the testing procedure. Based on
significance levels α = 0.10, 0.05, 0.01, compute the sample proportion of simulated
t-values that fell in the rejection region. The proportion of simulated rejected t-values
under the null is called the empirical size of a test. The three values should be close
to the actual α levels.
4