chapter 13: the simple linear regression model

Question 1

the simple linear regression model

Answer 1

the simple linear regression model assumes that the relationship between the dependent variable and independent variable can be approximated by a straight line

y: decedent variable
x: independent variable

Question 2

what can we use to tentatively decide wether there is an approximate straight line relationship between x and y

Answer 2

scatter plot

scatter diagram

Question 3

what is the the simple linear regression model formula

Answer 3

y = B0 + B1x + E

contains the mean level Uy

the y intercept B0

the slope B1

the error term E

Question 4

what is the mean level of the simple linear regression model formula?

Answer 4

Uy = B0 + B1x

the line of means

the values of y can be represented by the mean level

the value changes in the straight line represented by Uy

the y intercept: B0

the slope: B1

Question 5

the error term E

Answer 5

describes the effects on y of all factors other than the value of independent variable x

can be positive, negative or 0

Question 6

what does it mean for the error term E to be 0?

Answer 6

there is no difference between the mean level Uy and and just y

Question 7

what does it mean for the error term E to be bigger than 0?

Answer 7

the point will be above what is should be according to the Uy = B0 + B1x

it will be above than the corresponding x value

Question 8

what does it mean for the error term E to be lower than 0?

Answer 8

the point will be below what is should be according to the Uy = B0 + B1x

it will be lower than the corresponding x value

Question 9

what is the impact of B1 (the slope)

Answer 9

if B1 is positive, the regression line will go up

if B1 is negative, the regression line will go down

Question 10

what are the regression parameters

Answer 10

the y intercept B0

the slope B1

Question 11

true or false

we can reflect the changes made in the regression line as a change in the independent variable causing a change in the dependent variable

Answer 11

false

we can say the effect of the independent variable on the dependent variable

we can say that the two variables move together and that the independent variable contributes to information predicting the independent variable

Question 12

the least square line

Answer 12

the best visual estimated regression line

y^ = b0 + b1x

y^: the predicted value of y

b0: point estimate of y intercept BO
b1: point estimate of slope of Uy B1

Question 13

how is the predicted value of the dependent variable y found

Answer 13

yî = b0 + b1xi

b1 = SSxy / SSxx

b0 = y- - b1x-

Question 14

what is the residual of an observation?

Answer 14

yi - y^

yi: the observed y

Question 15

what is the experimental region

Answer 15

the range of previously observed population sizes

Question 16

the point prediction of an individual value

Answer 16

the point prediction of an individual value of the dependent variable when the value of the independent variable is X0

here we predict the error term to be 0

Question 17

simple coefficient of determination

Answer 17

a measure of potential selfness in the simple linear regression model

r^2 (r squared)

explained variation / total variation

r^2 always bigger than 0, but never bigger than 1

the closer it is to 1, the larger the proportion of the total variation that is explained by the simple linear regression model, the greater it can predict y

Question 18

how do you calculate the error of prediction in the simple coefficient determination?

Answer 18

yi - y-

y- (mean y), only works if we are not considering changes to x

yi - y^ if we are considering the changes to x

Question 19

what is the total variation?

Answer 19

the sum of squared prediction errors

this quantity measures the Toal amount of variation exhibited by the observed values of y

explained variation + unexplained variation

Question 20

what is the unexplained variation

Answer 20

another name of the SSE

the sum of squared prediction errors when we use the predictor variable x

quantity that measures the amount of variation in the values of y that is not explained by the predictor variable

total variation- explained variation

Question 21

explained variation

Answer 21

total variation - unexplained variation

Question 22

what is the best way to get prediction accuracy

Answer 22

by calculating a prediction interval

Question 23

the simple correlation coefficient r

Answer 23

a measure of correlation and strength of linear relationship between x and y

r = +sqrt(r^2) if b1 is positive

r = -sqrt(r^2) if b1 is negative

Question 24

why can r be negative but not r^2

Answer 24

cause bruv, something ^2 can’t be negative

r though, it can be negative

stays between -1 and 1

Question 25

what does it mean for x and y to be highly related and positively correlated

Answer 25

r is near 1

Question 26

what does it mean for x and y to be highly related and negatively correlated

Answer 26

r is near -1

Question 27

what are the regression assumptions?

Answer 27

1. at any given value of x, the population of potential error term values has a mean equal to 0 2. there is a constant variance assumption at any value of x, the population of potential error term values has a variance that does not depend on the value of x error term values per x values have equal variances 3. normality assumption 4. independence assumption

Question 28

what does it mean for different populations of potential error term values per corresponding values of to have equal variances?

Answer 28

at any value of x, the population of potential error term values has a variance that does not depend on the value of x

Question 29

whats the normality assumption (3) of the regression assumptions?

Answer 29

at any given value x, the population of error term values has a normal distribution

Question 30

whats the independence assumption (4) of the regression assumptions?

Answer 30

any one value of the error term e is statistically independent of any other value of E E of a certain y independent to any other E of another y they don't affect each other

Question 31

what do the overall regression assumptions mean?

Answer 31

for every value of x, the population of potential error term values is normally distributed the mean of the population of error terms is 0 the variance does not depend on the value of x

Question 32

why do we predict the mean of the population of errors terms to be 0?

Answer 32

because it has a 50% chance of being positive, and 50% chance of being negative

Question 33

what is the mean square error?

Answer 33

the point estimate of the variance

Question 34

what is the standard error

Answer 34

the point estimate of the standard deviation

Question 35

which is the best line to observe data? why?

Answer 35

the least square regression line It is the line minimizing the sum of the squared residuals

Question 36

why is it dangerous to extrapolate out of the experimental region?

Answer 36

because we do not know that x and y have a linear relationship outside the experimental region

Question 37

explain what is the total variation, explained variation, and unexplained variation

Answer 37

The total variation is the sum of the squared prediction errors when we do not use the predictor x The unexplained variation is the sum of the squared prediction errors when we do use the predictor x The explained variation measures the improvement in the fit when we do use the predictor x

Question 38

when is a simple linear regression model useful?

Answer 38

when there is a significant relationship between x and y

Question 39

how do we test the si significance of the relationship between x and y

Answer 39

with the null hypothesis (h0) being B1 = 0 this says there is no change in the mean value of y associated with an increase of x ha: B1 =/= 0

Question 40

if the regression assumptions hold, how many degrees of freedom does the t distribution have (cause you gotta use t distribution)

Answer 40

n - 2

Question 41

is there a difference between using a one sided or two sided critical value or p-value?

Answer 41

nah boy

Question 42

at what significance level do we see very strong evidence that the regression relationship is significant?

Answer 42

0.05 significance level

Question 43

at what significance level do we see strong evidence that the regression relationship is significant?

Answer 43

0.01 significance level

Question 44

how do we test the significance of the y intercept

Answer 44

H0: B0 = 0 Ha: B0 =/= 0

Question 45

how do we reject H0 in favor of Ha when testing the significance of the y intercept?

Answer 45

setting the probability of a type 1 error

Question 46

what is the f test

Answer 46

another way of checking the significance of the slope checking if H0: B1 = 0

Question 47

how do you do the f test?

Answer 47

F = explained variation / (unexplained variation) / (n - 2))

Question 48

what is the difference between a confidence interval and a regression interval?

Answer 48

A confidence interval is intended to capture the mean value of y based on standard error sy^ A prediction interval is intended to capture an individual observation of y based on standard rarer s(y - y^)

Question 49

what does the distance value measure

Answer 49

The distance between xo and 𝑥- xo: value of x corresponding to a a certain point estimate or point prediction x-: mean of x values

Question 50

how does the distance value affect confidence intervals

Answer 50

the bigger the distance value, the larger the confidence interval