Midterm 2023, Solutions

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Question 1

Q1:
> .08+.02
[1] 0.1

Q2:
> .02+.25+.32
[1] 0.59

Q3:
> .02/(.02+.08)
[1] 0.2

Q4:
> .32/(.32+.08)
[1] 0.8

Q5:
FALSE

Q6:

> .32/(.32+.25+.02)
[1] 0.5423729


Q7:
FALSE (X is not Bernoulli)

Q8:
TRUE (Yis Bernoulli)

Q9:
FALSE

Q10:

px = c(.1,.5,.4)
xv = 1:3
EX = sum(pv*xv)
EX

[1] 2.24

Q11:
print(xx)
print(px)
VX = sum((xx-EX)^2 * px)
print(VX)
1] 1 2 3
[1] 0.1 0.5 0.4
[1] 0.4136

Q12:
.4, intuitively, it is positive but not too big.


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Question 2

Q13
> .4*.9
[1] 0.36

Q14
> .6*.2
[1] 0.12

Q15
> .12+.36
[1] 0.48

Q16
> .36/.48
[1] 0.75


Q17
> (.4*.1)/(.4*.1 + .6*.8)
[1] 0.07692308

Q18
.9^2

Q19
.4*.9^2/(.4*.9^2 + .6*.2^2)

Q20
.4*(.1*.9)/(.4*(.1*.9) + .6*(.8*.2))
[1] 0.2727273


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Question 3

Q21
3 

Q22
3

Q23

(1- .68)/2 = .16
1-.16 = .84
> 1-pnorm(0,3,3)
[1] 0.8413447

Q24

> 1-pnorm(0,8,6)
[1] 0.9087888

Q25

> .15*3 + .85 *8
[1] 7.25

Q26

> .15^2 * 9 + .85^2 * 36 + .15*.85*2*(.45*3*6)
[1] 28.278

Q27

> .5 * 2 + .5 * 7.25
[1] 4.625

Q28

> .5*sqrt(28.278)
[1] 2.658853

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Question 4, Simple Linear Regression

Q29

> 13.706 + .86885*110
[1] 109.2795

Q30

> 109.3 + 2*10.78
[1] 130.86

Q31

> .86885 + 2*.03451
[1] 0.93787

Q32

reject, p-value is 0 from ouput

Q33

> t = (.86885-1)/.03451
> t
[1] -3.800348

reject => True

Q34

> sqrt(.7188)
[1] 0.8478207

Q35

same as x and y, .847

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Question 5, Simple Linear Regression Model

Q36

> 20+50*2
[1] 120

Q37

sd of errors = 15

Q38
E(P) = 20 + 50 E(S) + E(eps) = 20+50*2
120

Q39

> sqrt(50^2 * .2^2 + 15^2)
[1] 18.02776