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STAT 425 Exam

创建日期：2024-04-27 16:05:38

所属分类：统计学statistics

标签： STAT 425

Study Problems

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STAT 425 Exam 1 Study Problems

Exam problems are generally be shorter than homework problems and may involve short
answer conceptual questions, quick calculations and R code interpretation or debugging. It is
not a multiple choice exam, although some multiple choice questions are possible.
The sample problems below are to help you test yourself and practice solving.
Problems on the exam will generally have fewer parts to them than the ones
below. Do not expect the actual exam problems to be exactly like this set in
terms of range of coverage or length. Work on these various problems as a way
to solidify your understanding.
This is the unsolved version of the study problems. Solutions will be posted by Monday
morning.
1
1. Twenty chicks (baby chickens) were randomly assigned to receive one of two diets, A or B,
with 10 in each group. Consider the model
yi = β0 + β1xi + ei, i = 1, 2, 3, . . . , 20.
Here yi denotes the 14-day weight gain for the ith chick, and
xi =
−1, if chick i receives Diet A;1, if chick i receives Diet B.
The data are arranged so that Chick numbers 1 - 10 received Diet A and Chick numbers
11-20 received Diet B.
a) Calculate x¯ and Sxx for this design.
b) Show that βˆ1 the least squares estimate of β1 equals 12(y¯B − y¯A), where y¯A and y¯B are the
sample means for weight gain on Diet A and Diet B, respectively.
c) Suppose y¯A = 101.2, y¯B = 123.7 and
∑20
i=1(yi − yˆi)2 = 49.0. Calculate the value of the
t-statistic for testing the null hypothesis that β1 = 0.
2
2. We can rewrite the model from Problem 1 in matrix form as y = Xβ + e, where
y =

y1
y2
...
y19
y20
 X =

1 −1
1 −1
... ...
1 1
1 1
 β =
(
β0
β1
)
e =

e1
e2
...
e19
e20

a) Show that for this design the columns of X are orthogonal to each other.
b) Show that for this design cov(βˆ0, βˆ1) = 0.
c) Find the leverage of the first observation. Recall that hi = xTi (XTX)−1xi.
3
3. Consider a model of the form y = Xβ + e, where X is an n × p full rank matrix (its
columns are linearly independent), y and e are n × 1, and β is p × 1. The least squares
estimator βˆ solves the matrix equation
XT (y−Xβˆ) = 0.
Let yˆ = Xβˆ. Show or explain why each of the following equations holds, using the least
squares equation as a starting point:
a) yˆT (y− yˆ) = 0
b) yˆTy = yˆT yˆ
c) (y−Xβˆ)T (y−Xβˆ) = yTy− yˆTyˆ
d) RSS = ‖y− yˆ‖2 = yTy− βˆTXTXβˆ
e) Which, if any, of equations a), b), c), or d) says that the vector of residuals and vector of
fitted values are orthogonal to each other?
4
4. Consider a model of the form y = Xβ + e, where X is an n × p full rank matrix (its
columns are linearly independent), y and e are n× 1, and β is p× 1. Assume X is a fixed
(non-random) matrix, E(e) = 0, and cov(e) = σ2I. The least squares projection matrix H is
an n× n matrix, the “hat’ ’ matrix, of the form X(XTX)−1XT.
For each of the following statements, verify that it is true, or state why it is false:
a) yˆ = Xβˆ = Hy
b) βˆ = Hβ
c) HH = H
d) (I−H)X = 0
e) cov(y− yˆ) = σ2(I−H)
5
5. A study was conducted to compare antibiotic (drug) treatment with placebo (no drug) for
a certain disease. The variables are:
xi1 = Pretreatment condition score
xi2 =
1, Treated with drug;0, Treated with placebo (no drug).
yi = Post-Treatment condition score (condition after treatment)
The following model was fit using the lm function in R: yi = β0 + β1xi1 + β2xi2 + ei,
i = 1, 2, . . . , n, where the working assumptions are that the errors ei are independently
distributed as N(0, σ2). Some results are below.
##
## Call:
## lm(formula = PostTreatment ~ Pretreatment + Drug, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.4110 -2.3897 -0.5214 1.6708 8.5890
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.4429 2.4216 -0.183 0.8562
## Pretreatment 0.9878 0.1611 6.132 1.5e-06 ***
## Drug -3.3896 1.6100 -2.105 0.0447 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.931 on 27 degrees of freedom
## Multiple R-squared: 0.6762, Adjusted R-squared: 0.6522
## F-statistic: 28.19 on 2 and 27 DF, p-value: 2.446e-07
a) What was the overall sample size, n?
b) In the model summary, a t value is reported for Drug. State the null hypothesis and
alternative hypothesis for this test, expressed in terms of the mathematical parameters (β0,
β1, β2, σ2). Is th enull hypothesis rejected at level 0.05?
6
c) At the end of the model summary, an F test result is reported. State the null hypothesis
and alternative hypothesis for this test, expressed in terms of the mathematical parameters
(β0, β1, β2, σ2). Is the null hypothesis rejected at level 0.05?
d) Based on the fitted model, estimate the expected post-treatment condition score for
a new patient with pre-treatment score of 10 if they receive the drug. Also compute the
post-treatment condition score if they received the placebo.
e) What does the $fit component in the output below tell us? Explain it briefly.
predict(mod1, newdata=data.frame(Pretreatment=12, Drug=0),
se.fit=TRUE, interval="prediction", level=0.90)
## $fit
## fit lwr upr
## 1 11.41096 4.383872 18.43804
##
## $se.fit
## [1] 1.251574
##
## $df
## [1] 27
##
## $residual.scale
## [1] 3.931175
7
6. This problem considers the same data and model as in Problem 5.
a) A scatter plot of studentized residuals versus fitted values is shown below.
0 5 10 15 20
−
1
0
1
2
fitted(mod1)
rs
tu
de
nt
(m
od
1)
There is a hint of increasing vertical spread in the graph as we move from left to right. If that
effect is real, what does it suggest about the model assumptions we have made? Describe
briefly.
b) What are the values being graphed below, and what do they tell us?
library(faraway)
halfnorm(influence(mod1)$hat)
8
0.0 0.5 1.0 1.5 2.0
0.
00
0.
05
0.
10
0.
15
0.
20
Half−normal quantiles
So
rte
d
Da
ta
17
25
c) What are the values being graphed below, and what do they tell us?
halfnorm(cooks.distance(mod1))
0.0 0.5 1.0 1.5 2.0
0.
00
0.
05
0.
10
0.
15
0.
20
Half−normal quantiles
So
rte
d
Da
ta
23
30
9
7. Continuing with the data and variables defined in Problem 5, consider the following R
code and results:
library(MASS)
lambdas = boxcox(PostTreatment+1 ~ Pretreatment+Drug, data=df)
−2 −1 0 1 2
−
10
0
−
80
−
60
−
40
λ
lo
g−
Li
ke
lih
oo
d
95%
attributes(lambdas)
## $names
## [1] "x" "y"
lambdas$x[lambdas$y==max(lambdas$y)]
## [1] 0.4242424
a) Write out the class of regression models being fit here, expressed in terms of our original
variables xi1, xi2, yi, ei, and the parameters β0, β1, β2, σ2, and λ.
b) What is the estimated numerical value of λ? Also, give an approximate 95% confidence
interval for λ.
10
c) Do these results indicate that we should modify the linear model we fit in Problem 5? If
so what do you recommend?
11
8. Consider a model of the form yi = β0 + β1xi + xiei, 1, 2, . . . , n, where yi is the observed
response, the xi are observed, positive (>0), nonrandom values of an explanatory variable,
and the errors ei have mean zero, are uncorrelated and have constant variance σ2.
a) Find expressions for the expected value and variance of yi, i = 1, 2, . . . , n.
b) Find Cov(yi, yj) for i 6= j.
c) The optimal weighted least squares estimator for (β0, β1) is obtained by minimizing a
criterion of the form
RSSw(β0, β1) =
n∑
i=1
wi(yi − β0 − β1xi)2
as long as we chose the weights correctly. What are the optimal weights wi, i.e., what should
we plug in for wi?
d) Consider a constructed response variable zi = yi/xi. Show that zi follows a homoscedastic
(constant variance) linear model with a transformed explanatory variable, and write out the
form of the model.
e) Write out the form of the ordinary least squares RSS(β0, β1) for the model in d), and
show that it reduces to RSSw(β0, β1) in c) with the optimal of weights wi.

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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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