lecture 2

Question 1

derive this

Answer 1

Question 2

When do you apply the chain rule in differentiation?

Answer 2

You apply the chain rule whenever you are differentiating a composite function, meaning a function nested inside another function.

Question 3

What is the chain rule

Answer 3

Question 4

What do we differentiate?

Answer 4

Functions

Question 5

What does this mean?

Answer 5

Question 6

apply the chain rule to differentiate the composite function below (hint, identify the inner vs outer function)

Answer 6

Question 7

I want to differentiate with respect to X, how do i write this using proper notation?

Answer 7

Question 8

I want to differentiate with respect to t, how do i write this using proper notation? What about y?

Answer 8

Question 9

What is the correct notation for differentiating a function with multiple variables, how do i differentiate with respect to 1, x

e.g., f(x, y), and I want to differentiate with respect to 1

Answer 9

Question 10

Least squares estimation - what is this

Answer 10

They are mathematically derived formulas that minimize the sum of squared errors.

formulas are the least squares estimates for the coefficients 𝛽0 (intercept) and 𝛽1 (slope) in simple linear regression.

Question 11

What is the formula for the sum of squared errors

Answer 11

Question 12

Whats the difference between B0 and B1 and the below

Answer 12

Question 13

Whta is b0hat and b1hat least squares estimates of?

Answer 13

b0 and b1

Question 14

What do we mean by centroid of data?

Answer 14

Question 15

Sxx what is the formula for this and whats it reflecting

Answer 15

This sums up the squared differences between each participant’s x-score (x𝑖 ) and the mean of all x-scores (xˉ).

Interpretation:
*Large Sxx → More spread-out x-values (higher variance in x).
*Small Sxx → x-values are closer to the mean (lower variance in x).

Question 16

What is the value of B0(hat) that minimises thee SSE ?

Answer 16

Question 17

What is the value of B1(hat) that minimises thee SSE ?

Answer 17

Question 18

write mean of x (xbar) another way

Answer 18

Question 19

Syy what is the formula for this and whats it reflecting

Answer 19

This sums up the squared differences between each participant’s y-score (𝑦𝑖 ) and the mean of all
y-scores (𝑦ˉ).

Interpretation:
*Large Syy → More spread-out y-values (higher variance).
*Small Syy → y-values are closer to the mean (lower variance).

Question 20

Sxy - what is the formula for this and whats it reflecting

Answer 20

This represents the total covariation between x and y. It measures how much x and y change together.

Interpretation:
*Positive Sxy → x and y tend to increase together (positive correlation).
*Negative Sxy → When x increases, y tends to decrease (negative correlation).
*Near-zero Sxy → Little to no linear relationship between x and y.

Question 21

what does b1hat coefficient represent

Answer 21

the expected change in Y for a unit increase of X

Question 22

Can I apply the regression equation to predict any value of an iv for someone outside of the study

Answer 22

Not if values lie outside of the range of x-values used to fit the model: the linear relationship might not hold outside of this range. It can be dangerous to extrapolate

Question 23

what does goodness of fit measure

Answer 23

measures how widely scattered the points are across the line of bast fit

Question 24

How can we quantify the variation in data across the line of best fit ?

Answer 24

variance

Question 25

when adding an error term to the statistical model, what is the errors term expected value, variance, and correlation matrix?

Answer 25

In the model Ei is a random variable whos expected value is 0, with a variance of sigma squared. And the Ei's are all independent which is what meant by the third bullet point We assume these are independent and normally distributed

Question 26

Why do we need the error terms to be independent?

Answer 26

so we can say Yi is normally distributed

Question 27

For a standard regression equation with 1pv and 1 error term, what is random vs fixed variables?

Answer 27

Question 28

Whats the difference between OLS and MLE?

Answer 28

OLS (least squares estimates) estimates β0 and β1, but it does not explicitly estimate σ2. OLS is a specific case of Least Squares Estimation, and in simple linear regression, they are the same thing. (Minimizes squared errors) MLE estimates all three parameters: β0,β1 and σ2 (Maximizes likelihood). Maximizing the likelihood means choosing the values of β0, β1 and σ2 that make the observed data most probable.

Question 29

What is the mean and variance of:

Answer 29

Question 30

What sis the formula for the SSE (sum of squared errors)?

Answer 30

Question 31

Why Do We Differentiate the SSE with Respect to β0 and β1 in Least Squares Estimation?

Answer 31

because we are trying to find the values of β0 and β1 that minimize the SSE.