Chpt 13 - Simple Linear Regression

Question 1

What is the statistical method to model the linear relationship between 2 numerical variables?

Answer 1

Simple linear regression

Question 2

How do we know if two variables are linearly related?

Answer 2

If the mean of response variable Y is linearly dependent upon the value of predictor variable X

Question 3

What is the linear equation that determines a fitted line?

Answer 3

Also called the regression equation

Y = βo + β1x

Where:

βo - is the intercept
B1- is the slope

Question 4

What is a graphical method to determine the relationship between X (predictor) and Y (response)?

Answer 4

Scatter plots

Question 5

When looking at a scatter plot, if the means of Y at different values of X are close to a straight line (although not necessarily right on the line), what can be said about the relationship?

Answer 5

It is linearly related

Question 6

If the mean of Y decreases as the value of x increases, what type of linear relationship is this?

Answer 6

Negative

Question 7

If the mean of Y increases as the value of X increases, what type of linear relationship is this?

Answer 7

Positive

Question 8

What variable is X when looking at simple linear regression?

Answer 8

The predictor value

Question 9

What variable is Y when looking at simple linear regression?

Answer 9

The response value

Question 10

How do we denote the mean of Y when the predictor value is x?

Answer 10

μY|X=x

Question 11

What is the best fitted line that is found based on the least-squares criterion?

Answer 11

Regression line

or least-squares line

Question 12

What is the least squares criterion?

Answer 12

The line that best fits a set of data points is the one having the SMALLEST possible sum of the squared errors (residuals) which are made in using the fitted line to predict the y values

Basically, it helps us to determine the best line for the set of data points

Question 13

Which value represents the sum of square of the difference between x and it’s mean

Answer 13

Sxx

Question 14

What does Sxx represent?

Answer 14

The sum of the square of the difference between x and its mean

Question 15

What is the computing formula for Sxx?

Answer 15

Sxx = Σxsquared - ((Σx)squared/n)

Question 16

Which value represents the measure of the total variability of the yi’s from y?

Answer 16

Syy

Question 17

What does Syy represent?

Answer 17

The measure of the total variability of the yi’s from the y

Question 18

What is the computing formula for Syy?

Answer 18

Syy = Σysquared - ((Σy)squared/n)

Question 19

What value represents the sum of the product of the differences between x values and the mean of x and the differences between y values and the mean of y?

Answer 19

Sxy

Question 20

What does Sxy represent?

Answer 20

the sum of the product of the differences between x values and the mean of x and the differences between y values and the mean of y

Question 21

What is the computing formula for Sxy?

Answer 21

Sxy = Σxy - ((Σx)(Σy)/n)

Question 22

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. The values (all in $1000 except n) are:

Age Price ($1000)
1 14
1 13
3 13
4 10
4 10
5 9
5 9
6 7
7 7
7 8
8 7
8 6
10 5
10 4
13 3

What is the value of Σx?

Answer 22

Σx = 92

Question 23

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. The values (all in $1000 except n) are:

Age Price ($1000)
1 14
1 13
3 13
4 10
4 10
5 9
5 9
6 7
7 7
7 8
8 7
8 6
10 5
10 4
13 3

What is the value of Σxsquared?

Answer 23

Σxsquared = 724

Question 24

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. The values (all in $1000 except n) are:

Age Price ($1000)
1 14
1 13
3 13
4 10
4 10
5 9
5 9
6 7
7 7
7 8
8 7
8 6
10 5
10 4
13 3

What is the value of Σy?

Answer 24

Σy = 125

Question 25

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. The values (all in $1000 except n) are: Age Price ($1000) 1 14 1 13 3 13 4 10 4 10 5 9 5 9 6 7 7 7 7 8 8 7 8 6 10 5 10 4 13 3 What is the value of Σysquared?

Answer 25

Σysquared = 1193

Question 26

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. The values (all in $1000 except n) are: Age Price ($1000) 1 14 1 13 3 13 4 10 4 10 5 9 5 9 6 7 7 7 7 8 8 7 8 6 10 5 10 4 13 3 What is the value of Σxy?

Answer 26

Σxy = 616

Question 27

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. Important values (all in $1000 except n) are: n = 15 Σx = 92 Σxsquared = 724 Σy = 125 Σysquared = 1193 Σxy = 616 What is the value of Sxx?

Answer 27

Sxx = Σxsquared - ((Σx)squared/n) = 724 - (92squared/15) = 159.733

Question 28

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. Important values (all in $1000 except n) are: n = 15 Σx = 92 Σxsquared = 724 Σy = 125 Σysquared = 1193 Σxy = 616 What is the value of Syy?

Answer 28

Syy = Σysquared - ((Σy)squared/n) = 1193 - (125squared/15) = 151.333

Question 29

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. Important values (all in $1000 except n) are: n = 15 Σx = 92 Σxsquared = 724 Σy = 125 Σysquared = 1193 Σxy = 616 What is the value of Sxy?

Answer 29

Sxy = Σxy - ((Σx)(Σy)/n) = 616 - (92x125/15) = -150.667

Question 30

How do we calculate the value of slope?

Answer 30

b1 = Sxy/Sxx

Question 31

How do we calculate the value of the y intercept?

Answer 31

bo = ȳ - b1x̄

Question 32

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. Important values (all in $1000 except n) are: n = 15 Σx = 92 Σxsquared = 724 Σy = 125 Σysquared = 1193 Σxy = 616 Sxx = 159.733 Syy = 151.333 Sxy = -150.667 What is the value for the slope?

Answer 32

b1 = Sxy/Sxx = -150.667/159.733 = -0.9432

Question 33

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. Important values (all in $1000 except n) are: n = 15 Σx = 92 Σxsquared = 724 Σy = 125 Σysquared = 1193 Σxy = 616 Sxx = 159.733 Syy = 151.333 Sxy = -150.667 b1 = -0.9432 What is the value for the intercept?

Answer 33

bo = ȳ - b1x̄ = (Σy)/n - b1*(Σx)/n =(Σy - b1*Σx)/n = (125-(-0.9432)*92)/15 = 14.118

Question 34

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. Important values (all in $1000 except n) are: n = 15 Σx = 92 Σxsquared = 724 Σy = 125 Σysquared = 1193 Σxy = 616 Sxx = 159.733 Syy = 151.333 Sxy = -150.667 b1 = -0.9432 What will happen to the cost of a car when it becomes 1 year older?

Answer 34

It's price will decrease by 0.9432 thousand

Question 35

When we use the fitted regression equation to make a prediction, what should we avoid? What does this mean? Why do we avoid it?

Answer 35

We avoid extrapolation This means predicting the value of a response variable when the value of the predictor variable is outside the observed range. Because the very high and low ends are always going to give you really wonky numbers... like a car doesn't become worth a negative amount of money just because it is now 20 years old. Cheap yes, them pay you to take it away, not likely :)

Question 36

What is the correlation coefficient and what is it denoted by?

Answer 36

Denoted by r Measures the strength of the linear relationship between the response variable and predictor variable -1 is a strong negative relationship 1 is a strong positive relationship 0 is no relationship

Question 37

How to the values of the correlation coefficient tell us about the relationship between the response variable and predictor variable?

Answer 37

-1 is a strong negative relationship 1 is a strong positive relationship 0 is no relationship

Question 38

How is the correlation coefficient (r) calculated?

Answer 38

r = Sxy/(√SxxSyy)

Question 39

We are analyzing the relationship between the ages of used cars and sale prices and 15 cars are selected. Important values (all in $1000 except n) are: n = 15 Σx = 92 Σxsquared = 724 Σy = 125 Σysquared = 1193 Σxy = 616 Sxx = 159.733 Syy = 151.333 Sxy = -150.667 b1 = -0.9432 What is the correlation coefficient? What does this tell us about the relationship between age of the car and the sale price?

Answer 39

r = Sxy/√(SxxSyy) = -150.667 / √(159.733*151.333) = -0.969 There is a strong negative linear association so as the age increases, the sale price decreases

Question 40

What would an r=0.01 value tell us?

Answer 40

There is no linear relationship

Question 41

What would an r=0.5 value tell us?

Answer 41

That there is a positive linear relationship, but it's not very strong

Question 42

What would an r=1 value tell us?

Answer 42

A very strong positive linear relationship

Question 43

What would an r=-0.5 value tells us?

Answer 43

There is a negative linear relationship, but its not very strong

Question 44

What would an r=-1 value tell us?

Answer 44

A very strong negative linear relationship

Question 45

What is the coefficient of determination and what is it denoted by?

Answer 45

Denoted as Rsquared A method to evaluate the utility of a regression equation for making predictions. It measures the percentage of variation in the observed values of the response variable that can be explained by the regression model

Question 46

How can we calculate the coefficient of determination (Rsquared)?

Answer 46

Rsquared = rsquared The value is always between 0 and 1

Question 47

What does the value of the coefficient of determination (Rsquared) tell us?

Answer 47

The value is always between 0 and 1. When the Rsquared value is near 1, it indicates that the regression model is useful for making predictions

Question 48

What is the distribution of the response variable at a given predictor value called?

Answer 48

Conditional distribution because the distribution depends on the x value....this is the 3D model the guy drew with the superimposed distributions

Question 49

We have a correlation coefficient value of r=-0.9691, what is the coefficient of determination?

Answer 49

Rsquared = rsquared =(-0.9691)squared =0.9392 =93.92% of the variation in the observed y-values can be explained by the linear regression equations The larger the Rsquared value is, the more useful it is

Question 50

What is the mean and standard deviation of the conditional distribution called

Answer 50

Conditional mean and conditional standard deviation

Question 51

For a single, observation of the response variable, it is very unlikely to be the condiitonal mean exactly. How do we introduce this to the slope calculation?

Answer 51

y = βo + β1x + ε Where ε is the error term added to the model to capture the variation

Question 52

What assumptions need to be met for making inferences about a linear regression equation?

Answer 52

Normal population - each conditional distribution should be normal conditional mean of Y at X=x is βo + β1x Equal standard deviation of all the conditional standard deviations Independent observations

Question 53

What is a residual and what is it denoted by?

Answer 53

It is denoted by e It is the difference between an observation and where you expect it to be It is an estimate of error ε

Question 54

What is the equation for the residual?

Answer 54

ei = yi - (bo+b1xi)