Midterm

Question 1

Probability Density Function

Expected Value

Answer 1

Expected Value: E = ∑xf(x)

Question 2

Probability Density Function

Variance

Answer 2

σ² = E(x²) - (E(x))²

Question 3

Linear Transformation

Linear transformation

Answer 3

Y = a + bX

Question 4

Linear Transformation

Expected Value

Answer 4

μy = a + bμx

Question 5

Linear Transformation

Variance

Answer 5

σ² = b²σ²x

Question 6

Normal Distribution of X

Distribution

Answer 6

x ~ N(μ,σ²)

Question 7

Normal Distribution of X (population)

Z = ?

Answer 7

Z = (X - μ)/σ ~N(0,1)

Question 8

Sample normal distribution of X (sample)

Distribution

Answer 8

X̂ ~ N(μ,σ²/n)

Question 9

Sample normal distribution of X (sample)

Z

Answer 9

z = (X - μ)/ (σ/√n) ~ N(0,1)

Question 10

Binomial Probability

Probability of k successes

Answer 10

P(Y=k) = (n k)p^k(1-p)^n-k

Question 11

Binomial Probability

E(Y) =

Answer 11

E(Y) = np

Question 12

Binomial probability

V(Y) =

Answer 12

V(Y) = np(1-p)

Question 13

Binomial Probability

Distribution

Answer 13

X~Bin(n,p)

Question 14

Joint pdf

Covariance X,Y

Answer 14

σx,y = E(XY) - E(X)*E(Y)

Question 15

Joint pdf

E(XY)

Answer 15

E(XY) = ∑xyh(x,y)

Question 16

Joint pdf

correlation x,y

Answer 16

ρ = σx,y/σx*σy

Question 17

Linear combination W

W =

Answer 17

W = a + bX + cY

Question 18

Linear combination W

Expected value

Answer 18

E(w) = μw = a + bμx + cμy

Question 19

Linear combination W

Variance

Answer 19

v(x) = σ²w = b²σ²x +2bcσxy + c²σ²y

Question 20

Test procedure (σ known)

Z test

Answer 20

Z = (X - μ)/ (σ/√n)

Question 21

Test procedure (s known)

T test

Answer 21

T = (X - μ)/ (S/√n) ~t(n-1)

Question 22

Test procedure (s known)

CI T test

Answer 22

CI = X ∓ tα/2;n-1 * S/√n

Question 23

Proportion test

Answer 23

Z = P^ - p / √p(1-p)/n

Question 24

proportion test

CI

Answer 24

ci =p̂+- zα/2 * √p̂(1-p̂)/n

Question 25

Variance test

Answer 25

W = (n-1)S²/σ²

Question 26

Variance test | CI

Answer 26

``` L = (n-1)S²/xα/2;n-1 U = (n-1)S²/1-xα/2;n-1 ```

Question 27

Simple linear regression Basic assumption

Answer 27

E(y|x) = β0 + β1X

Question 28

Simple linear regression sample equation

Answer 28

ŷ = B0 + B1X

Question 29

Simple linear regression Covariance

Answer 29

Sx,y = 1/n-1 ( ∑XiYi - nXYhat) Also = rxy * Sx * Sy

Question 30

Simple linear regression Variance

Answer 30

Sx² = 1/n-1 (∑x²i) - n(x̄)²

Question 31

Simple linear regression Slope

Answer 31

B1 = Sx,y/S²x = (rxy*Sx*Sy)/ S²x

Question 32

Simple linear regression Intercept

Answer 32

B0 = Yhat - B1x̄

Question 33

Simple linear regression Residual

Answer 33

ei = Yi - Y^i

Question 34

Simple Linear Regression SSR

Answer 34

SSR = B1²(n-1)S²x

Question 35

Simple Linear Regression correlation

Answer 35

rx,y = Sx,y/Sx*Sy

Question 36

Simple linear regression SST =

Answer 36

SST = SSR + SSE = (n-1)Sy^2

Question 37

Simple linear regression MSE

Answer 37

MSE = SSE/n-2 = S²e

Question 38

Simple linear regression standard error slope

Answer 38

SB1 = Se/√(n-1)*S²x | where Se = √MSE

Question 39

Simple linear regression Regression test for slope

Answer 39

T = B1- β1/ SB1 ~ t(n-2)

Question 40

Simple linear regression Regression slope CI

Answer 40

CI = B1 +- tα/2;n-2*SB1

Question 41

Simple linear regression Coefficient of determination

Answer 41

R² = SSR/SST = 1 - SSE/SST R²adj = 1 - SSE/SST *n-1/n-k-1

Question 42

Uniform probability Distribution

Answer 42

X ~ U(α,β)

Question 43

Uniform probability E(X)

Answer 43

E(X) = a+b/2

Question 44

Uniform probability V(x)

Answer 44

V(X) = (b-a)²/12

Question 45

Uniform probability Probability P(X<=x)

Answer 45

P(X<=x) = x-a/range

Question 46

Confidence interval standard formula

Answer 46

CI = sample stat +- critical value α/2 * Standard error

Question 47

MSR =

Answer 47

MSR = SSR / df

Question 48

Type I error

Answer 48

Wrongfully reject H0

Question 49

Type II error

Answer 49

Wrongfully accept H0

Question 50

If the question is about means and you know the POPULATION standard deviation ?

Answer 50

Z test

Question 51

If the question is about means and you know the SAMPLE standard deviation ?

Answer 51

T test

Question 52

If the questions is about regression, use the test for the slope.

Answer 52

T = b1- beta1/ SB1 ~ t(n-2)

Question 53

Proportion test P^ =

Answer 53

P^ = x/n

Question 54

Expectation of sample x E(x̄) =

Answer 54

E(x̄) = μ

Question 55

Variance of sample x V(x̄) =

Answer 55

V(x̄t) = σ²/n

Question 56

Standard deviation of sample x SD(x̄) =

Answer 56

SD(x̄) = σ/√n