Linear Regression

Question 1

Regression Analysis uses a _______model to predict a ______variable (dv) by using one or more _______variables (iv).

Answer 1

Statistical

Response

Predictor

Question 2

In regression analysis, β₀ and β₁ are called_______

Answer 2

parameters

Question 3

What are the four steps of hypothesis testing?

Answer 3

Step 1:

one-sided: H₀<μ H_a≥μ(no linear association between x and y – not useful for predicting y)

two-sided: H₀=μ H_a≠μ

Step 2:

t=(x ̅-μ₀)/(s⁄ √n) with df=n-1
t*=b₁/s{b}

Step 3: t {1- α, n-1} OR t {1- α/2, n-1}

Step 4: If t ≥ +crit val or ≤ -crit val reject H₀

Question 4

What is the simple linear regression model?

Answer 4

Y=β₀+β₁X₁+ε

Question 5

In linear regression, E(ε)=

Answer 5

0

Question 6

In linear regression, σ² {ε}=

Answer 6

σ²

Question 7

In linear regression, ε’s are/are not correlated and have covariance of ___.

Answer 7

ε’s are uncorrelated and have covariance of 0

Question 8

Least Squares Estimates of betas _____ the sum

n

                              ∑   [y<sub>1</sub>-(β<sub>0</sub>-β<sub>1</sub>x<sub>i</sub>)]<sup>2</sup>

                            (i=1)

Answer 8

minimize

Question 9

Interpretation of β₁

Y=β₀+β₁X₁+ε

Answer 9

For each increase in x, there is an increase/decrease in y.

(e.g., For each add’l hour a student watches tv, he loses .2 GPA points)

Question 10

Interpretation of β₀

Y=β₀+β₁X₁+ε

Answer 10

The mean when x=0

(e.g., On average, first year students who don’t watch tv have a GPA of 3.9)

Question 11

y ̂ is the ____ regression line.

Answer 11

estimated

Question 12

b₁ and b₀ are estimates for

Answer 12

β₁ and β₀

Question 13

What is the equation for b₁

Answer 13

(ss_xy)/(ss_xx)

Question 14

What is the equation for b₀

Answer 14

y ̅ - b₁x ̅

Question 15

What is the equation for SS_xx

Answer 15

All of the following equations are equal

∑(x_i - x ̅ )²

(∑x _i²) - n(x ̅ )²

(n-1) s_x²

Question 16

SS_xx must be positive/negative.

Answer 16

positive

Question 17

What is the equation for SS_xy

Answer 17

∑(x_i - x ̅ ) (y - y ̅ )

(∑x_iy_i) - n(xy̅)

Question 18

When creating a table for an estimated regression line, which 5 columns should you include?

Answer 18

x_i | y_i | x_i² | y_i² | x_iy_i

Question 19

What is the equation for the residual ε_i

Answer 19

ε_i = y_i-E(y_i)

Question 20

What is the equation for the residual e_i

Answer 20

e_i = y_i - y ̂_i

Question 21

s² is the ________

Answer 21

sample variance

Question 22

What is the equation for s²

Answer 22

All of the equations below are equal

(∑(x_i-x ̅ )²) / (n-1)

SSE/(n-2)

MSE

Question 23

s is the ____________

Answer 23

sample standard deviation

Question 24

What is the equation for s

Answer 24

√MSE

√(SSE/(n-2))

Question 25

SSE is

Answer 25

The sum of the squared errors

Question 26

What is the equation for SSE

Answer 26

All of the equations below are equal

∑e_i²

∑(y_i - y ̂_i)²

ss_yy - b₁²ss_xx

Question 27

What does s²=.045 and s= .212 mean?

Answer 27

If the dist of GPA for ppl who watch x hrs of tv is approx. normal, then about 95% of them are expected to have GPAs within 2(.212) units of their simple linear reg model

Question 28

You should assume ____ for hypothesis testing and confidence intervals

Answer 28

normality

Question 29

b₁ and b₀ are _______ for β₁ and β₀

Answer 29

least squares estimators

Question 30

Why do you want to have a large range of data?

Answer 30

The more variation you have, the better estimate of the slope you can get..

Question 31

sampling distribution of __(b₁)_need to check this_?

Answer 31

has a t-distribution of n-2,

because we estimate b₀ and b₁

Question 32

What does it mean to have a 95% CI?

Answer 32

If we took 100 samples of size xx, we would expect 95% of tem to contain value β₁

Interpretation: 95% of all b₁’s will fall within this range

Question 33

What is Interval Estimation?

Answer 33

CI for mean of Y when x=x_h

Question 34

SSTo

Answer 34

the error/variation when not using any model at all; never changes when using a diff model or using new variables; total var around y ̅

Question 35

SSE

Answer 35

error/variation when using SLR; the variation in y not explained by using x; too high equals too much error

Question 36

SSR

Answer 36

The error left after fitting the model; the chunk of variation in y explained by using x (we want this to be large)

Question 37

What are the components of the ANOVA table?

Answer 37

Source of Variation SS df MS
Regression SSR ÷ 1 = MSR
Error SSE _ ÷ _ n-2 = MSE
Total SSTo n-1

Question 38

What does an F-test for model usefulness tell us

Answer 38

if R² is signif, but not if it is useful

Question 39

What are the four steps in conducting an F-test

Answer 39

Step 1:

two-sided: H₀:β₁=0 H_a: β₁ ≠ 0

Step 2: F*=MSR/MSE =SSR/MSE (all are always positive; want F* to be >1)

Step 3: F {1-α, 1, n-2} (*numerator df always 1 in SLR)

Step 4: if F* > F {1-α, 1, n-2}, we reject H₀and we have evidence that the SLR model is useful

• ****in SLR (one predictor variable), the t-test for β₁=0 is the same as the F-test
• ****In SLR only F* = t*² √(f_crit) = t_crit