Reading Quiz 15

Question 1

least-squares regression

Answer 1

fits a straight line to data in order to predict a response variable y from the explanatory variable x

Question 2

inference about regression conditions

Answer 2

1. the observations must be independent
2. the true relationship is linear
3. the standard deviation of the response about the true line is the same everywhere
4. the response varies normally about the true regression line

Question 3

the observations must be independent condition

Answer 3

in particular, repeated observations on the same individual are not allowed

Question 4

the true relationship is linear condition

Answer 4

we can’t observe the true regression line, so we will almost never see a perfect straight-line relationship in our data
look at the scatter plot to check that the overall pattern is roughly linear
a residual plot against x magnifies any unusual pattern

Question 5

the standard deviation of the response about the true line is the same everywhere condition

Answer 5

look at the residual plot
the scatter of the data points (the vertical distance) about the y=0 line should be roughly the same over the entire range of the data

Question 6

the response varies normally about the true regression line condition

Answer 6

the residuals estimate the deviations of the response from the true regression line, so they should follow a normal distribution
make a boxplot, histogram, or stemplot of the residuals and check for clear skewness or other major departures from normality
slight departures from normality do not greatly affect inference for regression, so they are allowed, particularly when we have many observations

Question 7

regression model

Answer 7

says that there is a true regression line μy = α + βx that describes how the mean response μy varies as x changes

Question 8

true regression line

Answer 8

μy = α + βx

describes how the mean response μy varies as x changes

Question 9

the observed response y for any fixed x has a normal

Answer 9

σ for any value of x

Question 10

the parameters of the regression model are

Answer 10

the intercept α estimated by a, the slope β estimated by b, and the standard deviation σ estimated by s

Question 11

the true slope β says how much

Answer 11

change in y when x increases by 1

Question 12

the standard deviation σ describes

Answer 12

how much variation there is in responses y when x is fixed

Question 13

to estimate σ

Answer 13

use the standard error about the line, s

Question 14

s

Answer 14

regression standard error
s= sqrt((Σresiduals^2)/(n-2)) = sqrt((Σ(y-yhat)^2)/(n-2))
sample standard deviation of the residuals
spread of data (measure of variability) around the least squares regression line
“typical” amount of prediction error when using a linear regression model to make predictions

Question 15

calculator compute s

Answer 15

enter data into L1 and L2

STATS, TESTS, LinRegTTest

Question 16

regression standard error has how many degrees of freedom

Answer 16

n - 2

all t procedures in regression inference have n-2 degrees of freedom

Question 17

inference for regression goal

Answer 17

predict behavior of y for given values of x

Question 18

inference for regression cont

Answer 18

there is an “on average” straight line relationship between y and x
saying μy moves along a straight line as explanatory variable x changes

Question 19

inference can be done in two ways:

Answer 19

confidence intervals for the slope and a significance test to test the hypothesis that the true slope is 0

Question 20

confidence intervals for the slope of the true regression line have the general form

Answer 20

b ± t*SEb.

Question 21

in practice, we use software to find

Answer 21

the slope b of the least-squares line and its standard error SEb

Question 22

formula for confidence interval for slope of true regression line

Answer 22

b ± t*SEb where SEb = s/(sqrt Σ(x-xbar)^2)

Question 23

some calculators have the program to calculate the confidence interval for the slope

Answer 23

stat, tests, linregtint
if you don’t have the program you can run a linregttest to obtain the b and t values, then calculate the SEb value knowing that t = (b - β)/SEb
then substitute that into the b ± tSEb form with the t value from the t distribution chart using df = n - 2

Question 24

t =

Answer 24

(b - β)/SEb

Question 25

to test the hypothesis that the true slope is zero, use the

Answer 25

t statistic
t = (b - β)/SEb
also given by software
t statistic is the standardized slope of the LSRL
stat, tests, linregttest (L1 and L2)

Question 26

regression output from statistical software usually gives

Answer 26

t and its two sided p-value

for a one-sided test, divide the p-value by 2

Question 27

the most common hypothesis is

Answer 27

Ho: β = 0
this says there is no true linear relationship between x and y
it also says that straight-line dependence on x has no value for predicting y
it also says that the population correlation between x and y is zero

Question 28

To review: we use least-squares regression to study the relation between a couple of variables, both of
which are (quantitative, categorical).

Answer 28

quantitative

Question 29

Before doing regressions to study the relationship between two quantitative variables, we should explore the data by examining a _______ and a __________.

Answer 29

scatterplot, residual plot

Question 30

The statistic that describes the strength of a linear relationship, that is the same whichever variable is thought of as the explanatory variable, and which has a familiar relationship to the percent of variance in one variable explained by the other, is the ______ ______.

Answer 30

A. correlation coefficient (or just, the correlation)

Question 31

What is a residual?

Answer 31

A. A residual is the vertical distance between the data point and the regression line, or y - y-hat.

Question 32

The r-squared value, which is part of the regression output, tells us how much of what is what?

Answer 32

A. How much of the variation in the y variable is accounted for by the linear relationship with x.

Question 33

Suppose we draw lots of samples and compute a regression line for each sample. The slope and intercept of each sample line estimates a true value. Thus the slope and intercept we obtain from our sample are _____ that estimate population ______.

Answer 33

statistics; parameters

Question 34

One of the conditions for regression inference is that for any fixed value of x, the response variable y varies according to a _____ distribution.

Answer 34

normal

Question 35

Another assumption for regression inference is that for any fixed value of x, the repeated responses y are ____ of each other.

Answer 35

independent

Question 36

Another assumption for regression inference is that the means of the sets of y-values for each x value have what relationship to the x values?

Answer 36

A. That the means of the y’s for each x are a linear function of x: mean for y’s = alpha + beta * x

Question 37

Another assumption for regression inference is that what measure of dispersion is equal for each value of x?

Answer 37

A. The standard deviation of the y’s for the various x values.

Question 38

True or False: the slope and intercept we obtain from the least squares regression for our sample are unbiased estimators, respectively, of the line connecting the population means for each of the x’s.

Answer 38

true

Question 39

What is the unbiased estimator for the standard deviation of the y values around the regression line (in other words, the standard deviation of the y values around the means of each of those values for each x)?

Answer 39

A. The statistic called s, which is the standard error, or the standard deviation of the residuals. .

Question 40

The statistic s represents the estimate of the standard deviation ____ in the regression model.

Answer 40

A.  (sigma)

Question 41

The parameter we are usually most interested in estimating from regression output is the (slope, y-
intercept) of the line.

Answer 41

slope

Question 42

What is the general form for a confidence interval for regression slope?

Answer 42

A. b plus or minus t*SEb

Question 43

The most commonly tested hypothesis about regressions is that Beta, the “Population slope,” is 0. Can you put this hypothesis in some other phrasings?

Answer 43

A. That the straight line dependence on x is of no value in predicting y. Or that the population correlation between x and y is 0. Or that there is no true linear relationship between x and y in the population.

Question 44

If you form the ratio of the slope obtained in your sample to the standard error of that slope, what is the sampling distribution of that statistic?

Answer 44

A. It’s distributed according to the t distribution, with n-2 degrees of freedom.

Question 45

Regression output usually gives a two-sided p value for the hypothesis test that the population slope is 0. How do you obtain a one-sided p-value for the same hypothesis?

Answer 45

A. Divide the two-sided p-value by two.

Question 46

Suppose that in a residual plot, the values are close to 0 when x is low, but the residuals get bigger and bigger in absolute value as the x values get greater. What condition of regression is violated in this circumstance?

Answer 46

A. The condition that the standard deviation of the response around the true line is the same everywhere.

Question 47

Someone examines a residual plot and a scatterplot and observes a curvilinear pattern. What condition of regression is being violated, and what should the researcher consider doing in order to correct this?

Answer 47

A. The condition violated is that the true relationship is linear. The researcher should consider transforming one or more of the variables.