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STA 463 Exam #2 Spring 2020

take home part 20 points

Instructions:

• These are all full-work problems, and point values are noted next to each part. In order to

receive credit for a problem, your solution must show sufficient details so that the grader can

determine how you obtained your answer. No work = no credit.

• Carry all computations to at least two decimal places. Only round the final answer. Do not

round during intermediate steps.

• The exam is open book/notes. But you should work independently on this exam. You cannot

discuss the exam questions with anyone, and cannot share the exam questions in any format

with anyone. Violations could result in a report of academic dishonesty issue and all the

consequences would follow.

• for graduate students, I will multiply your points in Q3 by 0.6 so that the total remains 20

points.

Q1. (4 points) A study was conducted to determine if there was an association between

the size (weight, in grams) of twenty-seven mice and four predictors: their sex (male or female,

“x1M”; x1M = 0 if the mouse is female, x1M = 1 if male) and three measures related to the size of

their features (occipital-incisor length, “x2”; orbital width, “x3”; skull height, “x4”; all measured

in millimeters). Below is some regression output for a model using the response variable and the

four predictors. Residual plots are also included.

----------------------------------------------------------------------------------------------------------------------

Call:

lm(formula = y ~ x1 + x2 + x3 + x4)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -45.2811 24.0279 -1.885 0.07278 .

x1M 0.9048 0.7112 1.272 0.21659

x2 2.7080 0.8033 3.371 0.00275 **

x3 -0.5901 1.7788 -0.332 0.74323

x4 0.7591 1.6671 0.455 0.65332

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 1.735 on 22 degrees of freedom

Multiple R-squared: 0.4121, Adjusted R-squared: 0.3053

F-statistic: 3.856 on 4 and 22 DF, p-value: 0.01603

------------------------------------------------------------------------------------------------------------------------

(a). (2 points.) Please explain the regression coefficient associated with x1M, 0.9048, in

the context of this problem.

(b). (2 points.) The R2 and adjusted R2 are not very close to each other. Why?

Q2. (6 points) Consider a dataset with a response and four predictors. The design matrix

for the first-order additive model is calculated. (X0X)

−1

, part of the hat matrix, and the vector of

estimated parameters are given below.

----------------------------------------------------------------------------------------------------------------------

> solve(t(X)%*%X)

x1 x2 x3 x4

1.259327e-01 -4.058861e-05 -7.821644e-04 -9.386430e-03 -1.048498e-02

x1 -4.058861e-05 3.749686e-08 -4.554027e-08 -5.376013e-06 -2.788214e-06

x2 -7.821644e-04 -4.554027e-08 6.111386e-05 7.457683e-05 -2.964554e-04

x3 -9.386430e-03 -5.376013e-06 7.457683e-05 5.374121e-03 -2.488392e-03

x4 -1.048498e-02 -2.788214e-06 -2.964554e-04 -2.488392e-03 4.158201e-02

H=X%*%solve(t(X)%*%X)%*%t(X)

2

> H[1:6,1:6]

[,1] [,2] [,3] [,4] [,5] [,6]

[1,] 0.07032303 0.03062687 0.04481705 0.06149829 0.03062320 0.03062320

[2,] 0.03062687 0.05189401 0.04550879 0.05210952 0.04872989 0.04872989

[3,] 0.04481705 0.04550879 0.04620762 0.05625619 0.04354044 0.04354044

[4,] 0.06149829 0.05210952 0.05625619 0.07156503 0.04969182 0.04969182

[5,] 0.03062320 0.04872989 0.04354044 0.04969182 0.04613626 0.04613626

[6,] 0.03062320 0.04872989 0.04354044 0.04969182 0.04613626 0.04613626

> B=solve(t(X)%*%X)%*%t(X)%*%y

> B

[,1]

30.8202702

x1 0.5846615

x2 -3.2686964

x3 22.3888892

x4 42.4382015

------------------------------------------------------------------------------------------------------------------------

(a). (2 points.) If MSE = 39, 880, what is the estimated variance of B4 (the parameter

estimate associated with x4)?

(b). (2 points.) If the appropriate t-multiplier is 1.98, what is the 95% confidence interval

for β4?

(c). (2 points.) What is the estimate of the covariance between the first and second

residual?

Q3. (10 points for undergrad; 6 points for grad) Consider a dataset consisting of a

random sample of livestock sales at Wapello Livestock Sales in Wapello, Iowa over several months

in 1999-2000. Suppose the response variable Y is the selling price of the cow in dollars, and that

the predictor variables were age of the animal in years (X1), weight of the animal in 100’s of pounds

(X2), and whether or not it is an Angus cow (X3 is 1 if Angus, 0 otherwise).

(a). (4 points.) Fit a multiple regression model with all three predictors (first order terms

only, no higher order terms, no interactions). Please interpret the estimated regression coefficient

associated with the weight predictor and find a 95% confidence interval for this parameter.

(b). (3 points.) Consider the model you fitted in (a). Please test H0 : β1 = β2 = β3 = 0

vs. HA : not all three β’s are equal to 0 using the ANOVA F test.

(c). (3 points.) Please fit a second multiple regression model with the three predictors, as

well as interaction terms between X3 and the other two predictors, i.e., X1 ∗ X3, X2 ∗ X3. what is

the fitted model for non-Angus cows?

Q4. (4 points, grad only) Suppose that the normal error simple linear regression model

is applicable, except that the error variance is not constant. In particular, the larger the fitted

value the larger the error variance. In this case, does β1 > 0 still imply that there is a positive,

linear relationship between X and Y ? Explain, briefly.

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