Download mean GPA - Pomona College

Math 158 – Linear Models
Spring 2016
Jo Hardin
iClicker Questions
Review
1. The p-value is defined as:
(a) The probability that H0 is true.
(b) The probability that H0 is true
given the observed data.
(c) The probability of observing the
data.
(d) The probability of observing the
data given H0 is true.
(e) I’ve never encountered the
definition of the p-value.
Review
2. In order to draw a causative
conclusion, we typically need:
(a) a randomization experimental
design
(b) a random sample
(c) a very large sample
(d) a good theory for why the
causation should exist
(e) I have no idea
Review
3. Power is:
(a) The probability that H0 is true.
(b) The probability that H0 is false.
(c) The probability of observing our
data.
(d) The probability of rejecting H0.
(e) I know power is good, but I can’t
remember why.
SLR
4. True or False:
The equation y = 5x + 2 represents
the equation of a line with slope 2.
(a) True.
(b) False.
SLR
5. True or False:
The lines
y = 3x + 6 and
y = -1/3x + 2
are perpendicular.
(a) True
(b) False
SLR
6. Does the equation 12x - 8y + 4 =
0 determine a linear function of x?
(a) It does determine a linear
function.
(b) It does not determine a linear
function.
SLR
7. If the null hypothesis (that β1=0) is
true, our SSE reduces to:
(a) ∑ (𝑌̂i - 𝑌̅)
2
(b) ∑ (Yi – 𝑌̂i)
(c) ∑ (Yi - 𝑌̅)
2
2
SLR
8. The regression technical conditions
include:
(a) The Y variable is normally
distributed
(b) The X variable is normally
distributed
(c) The residuals are normally
distributed
(d) The slope coefficient is normally
distributed
(e) The intercept coefficient is
normally distributed
SLR
9. What happens if the technical
assumptions are violated?
(a) The inference goes askew.
(b) The line doesn’t fit.
(c) The sample isn’t representative.
(d) It isn’t appropriate to draw a
causation conclusion.
SLR
10. What other approach could we use
to evaluate the relationship between
belonging to a Greek organization and
GPA?
Consider: 110 students: GPA &
46 in a Greek organization & 64 not.
(a) chi-squared test of independence
(b) t-test for comparing means of
independent groups
(c) t-test for comparing means of
dependent groups
(d) z-test for comparing means of
independent groups
(e) chi-squared test of goodness-of-fit
SLR
11. Assuming that the 2009 season is
representative of all MLB seasons, we
would like to test if these data provide
convincing evidence that the slope of
the regression line for predicting runs
from on-base plus slugging is different
than 0. What are the appropriate
hypotheses?
(a) H0: b0 = 0
Ha: b0 ≠ 0
(b) H0: b1 = 0
Ha: b1 ≠ 0
(c) H0: β0 = 0
Ha: β0 ≠ 0
(d) H0: β1 = 0
Ha: β1 ≠ 0
SLR
12. Which of the below is the correct
set of hypotheses and the p-value for
testing the slope of the relationship
between absences and GPA
Coefficients:
Estimate Std. Err t-val
(Inter)
3.47
0.42 8.35
Absences -0.016
0.01 -1.9
Pr(>|t|)
7.8e-09
0.068
(a) H0:b1=0 Ha:b1≠0 p-value=0.068
(b) H0:β1=0 Ha:β1 < 0
p-value = 0.068/2 = 0.034
(c) H0:β1=0 Ha:β1 ≠ 0
p-value = 0.068/2 = 0.034
(d) H0:b1=0 Ha:b1 < 0
p-value = 0.068/2 = 0.034
(e) H0:β1=0 Ha:β1 ≠ 0 p-value=7.8x10-9
SLR
13. We created a 95% confidence interval
for the mean GPA given 10 absences to
be (3.20, 3.42). What is the correct
interpretation?
(a) There is a 95% chance that the mean
GPA of students with 10 absences is
between 3.20 and 3.42.
(b) 95% of GPA averages (for students
with 10 absences) are between 3.20 and
3.42.
(c) 95% of GPAs (for students with 10
absences) are between 3.20 and 3.42.
(d) We are 95% confident that the true
mean GPA (for students with 10
absences) is between 3.20 and 3.42.
(e) 95% of our intervals will have a mean
GPA between 3.20 and 3.42.
SLR
14. We created a 95% prediction interval
for an individual GPA given 10 absences
to be (3, 3.62). What is the correct
interpretation?
(a) There is a 95% chance that the mean
GPA of students with 10 absences is
between 3 and 3.62.
(b) 95% of GPA averages (for students
with 10 absences) are between 3 and
3.62.
(c) 95% of GPAs (for students with 10
absences) are between 3 and 3.62.
(d) We are 95% confident that the true
mean GPA (for students with 10
absences) is between 3 and 3.62.
(e) 95% of our intervals will have a mean
GPA between 3 and 3.62.
SLR
15. A confidence interval for a mean
response at X:
(a) has a constant (width) for any
value of X.
(b) increases in width as X increases.
(c) decreases in width as X increases.
(d) increases in width as X gets
farther from 𝑋̅.
(e) decreases in width as X gets
farther from 𝑋̅.
SLR
16. A prediction interval for an
individual observation at X:
(a) has a constant (width) for any
value of X.
(b) increases in width as X increases.
(c) decreases in width as X increases.
(d) increases in width as X gets
farther from 𝑋̅.
(e) decreases in width as X gets
farther from 𝑋̅.
SLR
17. R2 for the regression line for
predicting runs from on-base plus
slugging is 91.31%.
Which of the below is the correct
interpretation of this value?
91.31% of
(a) runs can be accurately predicted
by on-base plus slugging.
(b) variability in predictions of runs is
explained by on-base plus slugging.
(c) variability in predictions of onbase plus slugging is explained by
runs.
(d) variability in runs is explained by
on-base plus slugging.
(e) variability in on-base plus slugging
is explained by runs.
SLR
18. Which SLR assumption is violated?
(a) linearity
(b) constant variance of errors
(c) independent errors
(d) normal errors
(e) outliers
SLR
19. Which SLR assumption is violated?
(a) linearity
(b) constant errors
(c) independent errors
(d) normal errors
(e) outliers
SLR
20. Which residuals are “best” ?
SLR
21. Why do we plot the residuals
versus fitted instead of
explanatory?
(a) because the residuals are
correlated with the fitted variable.
(b) because the residuals are
correlated with the explanatory
variable
(c) because we want to be able to
extend the model to incorporate more
variables
(d) because the explanatory variable
is a linear combination of the fitted
values
SLR
22. Why do we plot the residuals
versus fitted instead of observed?
(a) because the residuals are
correlated with the fitted variable.
(b) because the residuals are
correlated with the observed variable
(c) because we want to be able to
extend the model to incorporate more
variables
(d) because the observed variable is a
linear combination of the fitted values
SLR
23. Consider the following regression
model:
𝐸 [𝑌] = 𝛽0 + 𝛽1 ∗ 𝑋
If we go from X=20 to X = 40 in the
model, we can interpret 𝛽1 as
(a) E[Y] is larger by an amount of
20𝛽1
(b) median[Y] is larger by a factor of
𝑒 20𝛽1 .
(c) E[Y] is larger by an amount of
𝛽1 ln2.
(d) median[Y] is larger by a factor
of 2𝛽1 .
SLR
24. Consider the following regression
model:
𝐸 [ln(𝑌)] = 𝛽0 + 𝛽1 𝑋
If we go from X=20 to X = 40 in the
model, we can interpret 𝛽1 as
(a) E[Y] is larger by an amount of
20𝛽1
(b) median[Y] is larger by a factor of
𝑒 20𝛽1 .
(c) E[Y] is larger by an amount of
𝛽1 ln2.
(d) median[Y] is larger by a factor
of 2𝛽1 .
SLR
25. Consider the following regression
model:
𝐸 [ln(𝑌)] = 𝛽0 + 𝛽1 ln(𝑋)
If we go from X=20 to X = 40 in the
model, we can interpret 𝛽1 as
(a) E[Y] is larger by an amount of
20𝛽1
(b) median[Y] is larger by a factor of
𝑒 20𝛽1 .
(c) E[Y] is larger by an amount of
𝛽1 ln2.
(d) median[Y] is larger by a factor
of 2𝛽1 .
SLR
26. Consider the following regression
model:
𝐸 [𝑌] = 𝛽0 + 𝛽1 ∗ ln(𝑋)
If we go from X=20 to X = 40 in the
model, we can interpret 𝛽1 as
(a) E[Y] is larger by an amount of
20𝛽1
(b) median[Y] is larger by a factor of
𝑒 20𝛽1 .
(c) E[Y] is larger by an amount of
𝛽1 ln2.
(d) median[Y] is larger by a factor
of 2𝛽1 .
SLR
27. The intercept in the multiple
regression model
a. should be excluded if one
explanatory variable has negative
values.
b. determines the height of the
regression line.
c. should be excluded because the
population regression function does
not go through the origin.
d.is statistically significant if it is
larger than 1.96.
SLR
28. The local utility company surveys
101 randomly selected customers. For
each survey participant, the company
collects the following: annual electric
bill ($) and home size (sq ft).
Regression equation:
Annual bill = 0.55 * Home size + 15
Predictor
Coef
SE Coef
T
P
Constant
15
3
5.0
0.00
Home size
0.55
0.24
2.29
0.024
What is the 99% confidence interval
for the slope of the regression line?
(A) 0.25 to 0.85
(B) 0.02 to 1.08
(C) -0.08 to 1.18
(D) 0.20 to 1.30
(E) 0.30 to 1.40
SLR
29. The confidence interval for a single
coefficient in a multiple regression
a. makes little sense because the
population parameter is unknown.
b. should not be computed because
there are other coefficients present in
the model.
c. contains information from other
hypothesis tests (here only 𝛽0 ).
d. should only be calculated if the
regression R2 is large.
Linear Algebra
2
30. Calculate [
−3
3 −1
(a) [
]
−2 2
0 −2
(b) [
]
2 5
0 0
(c) [
]
−6 2
0 0
][
1 2
−1
]
2
(d) None of the above
(e) This matrix multiplication is
impossible
Linear Algebra
31. If A and B are both 2x3 matrices,
which of the following is not defined?
(a) A+B
(b) AtB
(c) BA
(d) ABt
(e) More than one of the above
Linear Algebra
32. If A =
3
B = [1
3
AB?
(a) 0
(b) 1
(c) 3
(d) 4
(e) 8
0
2
1
2
3 1
[ 0 −1 3] and
−2 0 4
2
−1], what is the (3,2)-entry of
0
Multiple Linear Regression
33. Which of the below correctly
describes the roles of variables in this
regression model?
Est
SE
t value Pr(>|t|)
(Intercept) 197.96
59.20
3.34
0.0058
volume
0.71
0.06
11.67
0.0000
coverpb
-184.05
40.49
-4.55
0.0007
(a) response: weight
explanatory: volume, paperback cover
(b) response: weight
explanatory: volume, hardcover cover
(c) response: volume
explanatory: weight, cover type
(d) response: weight
explanatory: volume, cover type
MLR
34. An econometrician is interested in evaluating
the relation of demand for building materials to
mortgage rates in LA and SF.
Y = 10 + 5X1 + 8X2
where
X1 = mortgage rate in %
X2 = 1 if SF, 0 if LA
Y = demand in $100 per capita
holding constant the effect of city,
each additional increase of 1% in the
mortgage rate would lead to an
estimated average ________ in the
mean demand.
(a) predicted $500 more per capita
(b) predicted $500 less per capita
(c) predicted $5 more per capita
(d) predicted $5 less per capita
MLR
35. Referring to
Y = 10 + 5X1 + 8X2
where
X1 = mortgage rate in %
X2 = 1 if SF, 0 if LA
Y = demand in $100 per capita
the effect of living in LA rather than
SF is a ________ demand by an
estimated ________ holding the
effect of mortgage rate constant.
(a) larger; $800 per capita
(b) smaller; $800 per capita
(c) larger, $8 per capita
(d) smaller, $8 per capita
MLR
36. Referring to
Y = 10 + 5X1 + 8X2
where
X1 = mortgage rate in %
X2 = 1 if SF, 0 if LA
Y = demand in $100 per capita
the fitted model for predicting
demand in LA is ________.
(a) 10 + 5X1
(b) 10 + 13X1
(c) 15 + 8X2
(d) 18 + 5X1
MLR
37. Referring to
Y = 10 + 5X1 + 8X2
where
X1 = mortgage rate in %
X2 = 1 if SF, 0 if LA
Y = demand in $100 per capita
the fitted model for predicting
demand in SF is ________.
(a) 10 + 5X1
(b) 10 + 13X1
(c) 15 + 8X2
(d) 18 + 5X1
MLR
38. A dummy variable (0,1) is used as
an explanatory variable in a
regression model when:
(a) the variable involved is numerical.
(b) the variable involved is
categorical.
(c) a quadratic relationship is
suspected.
(d) when two explanatory variables
interact.
MLR
39. If a categorical explanatory
variable contains three categories,
then _________ dummy variable(s)
will be needed to uniquely represent
these categories.
(a) 1
(b) 2
(c) 3
(d) 4
MLR
40. An interaction term in a multiple
regression model may be used when:
(a) the coefficient of determination is
small.
(b) there is a quadratic relationship
between the response and
explanatory variables.
(c) neither one of two explanatory
variables contribute significantly to
the regression model.
(d) the relationship between X1 and Y
changes for differing values of X2.
MLR
41. Referring to
Y = 10 + 5X1 + 8X2
where
X1 = mortgage rate in %
X2 = 1 if SF, 0 if LA
Y = demand in $100 per capita
to test whether there is a location
effect on demand, one would use:
(a) an F test on the significance of the
whole regression model.
(b) a t test on the significance of β1.
(c) a t test on the significance of β2.
(d) None of the above.
MLR
42. The F statistic for testing the
entire regression model can be
expressed as:
(a) SSR/SSE.
(b) MSE/MSR.
(c) MSR/MSE.
(d) MSR/SST.
MLR
43. The adjusted R2 is "adjusted for"
the:
(a) number of predictors only.
(b) sample size only.
(c) number of predictors and the
sample size.
(d) None of the above.
MLR
44. In a multiple regression model,
which of the following is correct
regarding the value of the adjusted
R2?
(a) It can be negative.
(b) It has to be positive.
(c) It has to be larger than the
coefficient of multiple determination
(R2).
(d) It can be larger than 1.
MLR
45. Which of the following is NOT an
assumption for the multiple regression
model?
(a) Positive autocorrelation of error
terms.
(b) Normality of error terms.
(c) Independence of error terms.
(d) Constant variation of error terms.
(e) At any combination of values of
the independent variables, the error
terms have a mean of 0.
MLR
46. It is often a good idea to compute
R2adj, the adjusted R2, rather than just
R2. This is done in order to
(a) avoid overestimating the
importance of the independent
variables.
(b) correct any problems that may
arise from any of the model
assumptions being violated.
(c) avoid having a multiple regression
model that contains too many
variables.
(d) avoid conducting t tests for each
independent variable.
MLR
47. An application of the multiple
regression model generated the
following results involving the F test
of the overall regression model:
p-value = 0.0012, R2 = 0.67, s = 0.076.
The null hypothesis, which states that
none of the explanatory variables are
significantly related to the response
variable, should be rejected, at the
0.05 level of significance.
A) True
B) False
MLR
48.
𝐸[𝑌] = 𝛽0 + 𝛽1 𝑋1 + 𝛽2 𝑋2 + 𝛽3 𝑋1 2 + 𝛽4 𝑋22
Which test should be used to test the
significance of the higher ordered
terms (𝑋1 2 and 𝑋22 )?
H0: 𝛽3 = 𝛽4 = 0
Ha: At least one of 𝛽3 and 𝛽4 does not
equal 0.
A) the overall nested F test.
B) the R2
C) the partial nested F test.
D) the t test.
MLR
49. In a nested F test, the reduced
model corresponds to the null
hypothesis being true.
(a) True
(b) False
MLR
50. In a nested F test, the full model
corresponds to the alternative
hypothesis being true.
(a) True
(b) False
MLR
51. Each of the following linear
hypothesis can be tested using the
nested F-test with the exception of
(a) β2 = 1 and β3 = β4 / β5
(b) β2 = 0
(c) β1 + β2 = 1 and β3 = -2β4
(d) β0 = β1 and β1 = 0
MLR
52. A multiple regression analysis was
conducted for a dependent variable Y on
100 independent variables X1, X2, …, X100.
Among other things, R2 was computed to
be 0.87. What is the most appropriate
way to interpret this?
(a) 87 of the independent variables in the
model are capable of accurately
predicting Y.
(b) We will accurately predict Y 87% of
the time.
(c) 87 of the independent variables are
statistically significant, while the
remaining 13 variables should be
removed from the model.
(d) 87% of the variation in the observed
values y can be explained by the
independent variables.
MLR
53. Based on data from 9 days, a multiple
regression was fit to the Dow Jones Index
on the following 7 explanatory variables:
i. high temp
ii. low temp
iii. 1 if sunny
iv. 1 if Yankees won
v. # runs Yankees
scored
vi. 1 if Mets won
vii. # runs Mets scored
What do you think of R2?
(a) It will be low because none of the
variables are likely to predict the Dow.
(b) It will be high because that combo of
variables is likely to predict the Dow.
(c) It will be low for other (mathematical)
reasons.
(d) It will be high for other
(mathematical) reasons.
MLR
54. In case of multiple regression the
_______________ is the proportion
of variation in the response variable
that is explained by the combination
of explanatory variables.
(a) coefficient of correlation
(b) coefficient of partial determination
(c) coefficient of regression
(d) coefficient of multiple determination
MLR
55. The _______________measures
the proportion of variation in the
response variable that is explained by
each explanatory variable holding all
other explanatory variables constant.
(a) coefficient of correlation
(b) coefficient of partial determination
(c) coefficient of regression
(d) coefficient of multiple determination
MLR
56. For a multiple regression model,
the computer output shows that the
simple correlation coefficient between
the response variable and one of the
explanatory variables is 0.99. This
result indicates that most likely the
problem of multicollinearity exists in
this model.
(a) TRUE
(b) FALSE
MLR
57. Suppose that in a multiple
regression the F is significant, but
none of the t-ratios are significant.
This means that:
(a) multicollinearity may be present
(b) residuals are not independent
(c) the regression is good
(d) a nonlinear model would be a
better fit
(e) none of the above
MLR
58. All of the following are possible
effects of multicollinearity EXCEPT:
a) the variances of regression
coefficients estimators may be larger
than expected
b) the signs of the regression
coefficients may be opposite of what
is expected
c) a significant F ratio may result
even though the t ratios are not
significant
d) removal of one data point may
cause large changes in the
coefficient estimates
e) the VIF is zero
MLR
59. For a quiz on 100 topics(you
know nothing):
Kelly knows 85 topics.
Jamie knows 75 topics.
Parker knows 55 topics.
Riley knows 45 topics.
Who should you choose to help you
answer the questions?
(a) Kelly
(b) Jamie
(c) Parker
(d) Riley
(e) can’t tell
MLR
60. Who do you want to choose next?
Kelly knows 85 topics.
Jamie knows 75 topics.
Parker knows 55 topics.
Riley knows 45 topics.
(a) Jamie
(b) Parker
(c) Riley
(d) depends on overlap with Kelly
(e) depends on overlap with Jamie,
Parker, and Riley
MLR
61. If you can pick two people, who
do you pick?
Kelly knows 85 topics.
Jamie knows 75 topics.
Parker knows 55 topics.
Riley knows 45 topics.
(a) Kelly plus person who overlaps
least
(b) Kelly plus person who overlaps the
most
(c) The two with the least overlap
(d) The two with the most overlap
(e) The two who have the largest
union.
p-values
RA Fisher (1929)
“… An observation is judged significant, if it would rarely
have been produced, in the absence of a real cause of the
kind we are seeking. It is a common practice to judge a
result significant, if it is of such a magnitude that it would
have been produced by chance not more frequently than
once in twenty trials. This is an arbitrary, but convenient,
level of significance for the practical investigator, but it
does not mean that he allows himself to be deceived once
in every twenty experiments. The test of significance only
tells him what to ignore, namely all experiments in which
significant results are not obtained. He should only claim
that a phenomenon is experimentally demonstrable when
he knows how to design an experiment so that it will
rarely fail to give a significant result. Consequently,
isolated significant results which he does not know how to
reproduce are left in suspense pending further
investigation.”
George Cobb (2014)
Q: Why do so many colleges and
grad schools teach p = .05?
A: Because that's still what the
scientific community and journal
editors use.
Q: Why do so many people still use p
= 0.05?
A: Because that's what they were
taught in college or grad school.
Basic and Applied Social Psychology
(2015)
With the banning of the NHSTP (null hypothesis
significance testing procedures) from BASP, what
are the implications for authors?
Question 3. Are any inferential statistical
procedures required?
Answer to Question 3. No, because the state of
the art remains uncertain. However, BASP will
require strong descriptive statistics, including
effect sizes. We also encourage the presentation of
frequency or distributional data when this is
feasible. Finally, we encourage the use of larger
sample sizes than is typical in much psychology
research, because as the sample size increases,
descriptive statistics become increasingly stable
and sampling error is less of a problem. However,
we will stop short of requiring particular sample
sizes, because it is possible to imagine
circumstances where more typical sample sizes
might be justifiable.
American Statistical Association’s
Statement on p-values (2016)
1. P-values can indicate how
incompatible the data are with a
specified statistical model.
(a) TRUE
(b) FALSE
2. P-values do not measure the
probability that the studied hypothesis
is true, or the probability that the
data were produced by random
chance alone.
(a) TRUE
(b) FALSE
3. Scientific conclusions and business
or policy decisions should not be
based only on whether a p- value
passes a specific threshold.
(a) TRUE
(b) FALSE
4. Proper inference requires full
reporting and transparency.
(a) TRUE
(b) FALSE
5. A p-value, or statistical
significance, does not measure the
size of an effect or the importance of
a result.
(a) TRUE
(b) FALSE
6. By itself, a p-value does not
provide a good measure of evidence
regarding a model or hypothesis.
(a) TRUE
(b) FALSE
Dance of the p-values:
https://www.youtube.com/watch?v=5
OL1RqHrZQ8
Your own p-value:
https://www.openintro.org/stat/why0
5.php?stat_book=os
MLR
62. Which of following is true?
(a) Influential points always reduce
R2.
(b) High leverage points always
reduce R2.
(c) All outliers are influential points.
(d) When the data set includes an
influential point, the relationship
between x and y is nonlinear.
(e) None of the above.
MLR
63. Which of the below best describes
the outlier?
(a) influential
(b) low leverage
(c) high leverage
(d) none of the above
MLR
64. Which of the below best describes
the outlier?
(a) influential
(b) low leverage
(c) high leverage
(d) none of the above
MLR
65. 1 obs in top left, 25 each in
bottom right. r (correlation) is:
(a) 0.9-0.99
(b) 0.7-0.89
(c) 0.4-0.69
(d) 0 – 0.39
(e) < 0
MLR
65. Why does a case with large
leverage have only the potential to be
influential?
(a) If the sample size is large, all
outliers will be mitigated.
(b) If the response at that point is
consistent with the model given by
the other points, the model won’t
change.
(c) It may influence the coefficients
but not the residuals.
(d) It doesn’t, high leverage points
will always be influential.
MLR
66. To check whether the 𝜖i have
homogeneous variance:
(a) leverage plot
(b) Cook’s Distance plot
(c) DFBETAS plot
(d) Residual plot
(e) VIF plot
MLR
67. To check whether the regression
is being unduly influenced by the 11th
observation.
(a) leverage plot
(b) Cook’s Distance plot
(c) DFBETAS plot
(d) Residual plot
(e) VIF plot
MLR
68. To check whether the regression
on X3 is really linear (as the model
states).
(a) leverage plot
(b) Cook’s Distance plot
(c) DFBETAS plot
(d) Residual plot
(e) VIF plot
MLR
69. Is there an observation that does
not seem to fit the model?
(a) leverage plot
(b) Cook’s Distance plot
(c) DFBETAS plot
(d) Residual plot
(e) VIF plot
Shrinkage Methods
70. Recall that n is the size of the
data set and p is the dimension of the
coefficient vector.
What is the size of the matrix that
gets inverted in ridge regression?
(a) p x p
(b) n x n
(c) np x np
(d) n2 x n2
Shrinkage Methods
71. As λ  0 (in ridge regression or
lasso)
(a) RR coefficients  ∞
(b) RR coefficients  0
(c) RR coefficients  OLS coefficients
Shrinkage Methods
72. As λ  ∞ (in ridge regression or
lasso)
(a) RR coefficients  ∞
(b) RR coefficients  0
(c) RR coefficients  OLS coefficients
Shrinkage Methods
73. The main motivation of ridge
regression is that it:
(a) minimizes bias
(b) minimizes variance
(c) minimizes bias and variance
(d) maximizes bias
(e) maximizes variance
Shrinkage Methods
74. The best procedure to test for
multicollinearity is
(a) to examine the correlation matrix.
(b) to examine the pairs plot
(c) to use the variance inflation factor.
(d) to use ridge regression.
Shrinkage Methods
75. The lasso outperforms ridge
regression at the cost of an increase
in complexity.
(a) Always
(b) Sometimes
(c) Never
Shrinkage Methods
76. Which of the following is
associated with the procedure to get
regression parameter estimates with
smaller standard errors
(a) piece-wise regression
(b) polynomial regression
(c) ridge regression
(d) stepwise regression
(e) lasso regression
Smoothing
77. With step basis function, C0
through CK,
(a) all K+1 functions can be used as
explanatory variables.
(b) any K functions can be used as
explanatory variables
(c) C1 through CK should be used as
explanatory variables
(d) any K-1 functions can be used as
explanatory variables
(e) C2 through CK should be used as
explanatory variables
Smoothing
78. In the step function model, how is
𝛽𝑗 interpreted? (On the basis function
I(cj <= X < cj+1) .)
(a) the value of Y over cj <= X < cj+1
(b) the average value of Y over
cj <= X < cj+1
(c) the increase in Y from X < c1 to
cj <= X < cj+1
(d) the increase in average Y from X
< c1 to cj <= X < cj+1
Smoothing
79. In a polynomial model, what is
the jth basis function?
(a) X
(b) X2
(c) Xj-1
(d) Xj
(e) Xj+1
Smoothing
80. What mathematics gives the
needed information in order to
perform inference for any basis
function model?
(a) calculus
(b) Lagrange multipliers
(c) linear algebra
Smoothing
81. Fitting separate polynomial
models locally can be problematic
because
(a) the variability at the extremes is
high
(b) the higher order derivatives may
not be continuous
(c) the step function may not be
continuous
(d) the model can be numerically
unstable because the explanatory
variables are highly correlated
Smoothing
82. T o fit a piecewise linear function
with 3 knots, continuous at the nodes,
one needs
(A) 3 bases functions
(B) 4 bases functions
(C) 5 bases functions
(D) 6 bases functions
Smoothing
83. How many degrees of freedom do
we have left when fitting a cubic
regression spline with K knots?
(a) K
(b) K + 3
(c) K + 4
(d) n – K
(e) n – K – 4
Smoothing
84. Why is it difficult to extend local
regression (loess) to higher
dimension?
(a) in high dimensions all points are
far from one another
(b) inverting matrices with p > 2 is
computationally difficult
(c) distance is difficult to compute in
high dimensions
Smoothing
85. Tricubic weight functions:
(a) give decreasing (non-zero)
weights for all Xi values
(b) cannot be computed for s > 1
“proportion” of the data
(c) can only be computed using
Euclidean distance
(d) can sometimes be negative
(e) none of the above
ANOVA
86. Which of the following is a necessary
assumption to conduct a one-way ANOVA
comparing r population means?
(a) The r populations have equal
variances.
(b) The r populations all have normal
distributions.
(c) The samples are randomly selected,
independent samples.
(d) All of the above.
ANOVA
87. In a one-way ANOVA F test, the
"between group" variation is attributable
to:
(a)
experimental error.
(b)
unexplained variation.
(c)
treatment effects.
(d)
residual variation
ANOVA
88. Which of the following components in
an ANOVA table are not additive?
(a)
Sum of squares.
(b)
Degrees of freedom.
(c)
Mean squares.
(d)
It is not possible to tell.
ANOVA
89. The ______ sum of squares
measures the variability of the observed
values of the response variable around
their respective treatment means.
(a)
treatment
(b)
error
(c)
interaction
(d)
total
ANOVA
90. The ________ sum of squares
measures the variability of the sample
treatment means around the overall
mean.
(a)
treatment
(b)
error
(c)
interaction
(d)
total
ANOVA
91. What are the correct hypotheses for
testing for a difference between the mean
concentrations among the three levels?
(a)
H0: µ1=µ2=µ3
Ha: µ1≠µ2≠µ3
(b)
H0: µ1≠µ2≠µ3
Ha: µ1=µ2=µ3
(c)
H0: µ1=µ2=µ3
Ha: at least one pair of means are
different
(d)
H0: µ1=µ2=µ3=0
Ha: at least one pair of means are
different
(e)
H0: µ1=µ2=µ3
Ha: µ1>µ2>µ3
ANOVA
92. When conducting a one-way
ANOVA, the ______ the betweentreatment variability is when
compared to the within-treatment
variability, the ________ the value of
F will be tend to be.
(a)
smaller, larger
(b)
larger, smaller
(c)
larger, larger
(d)
smaller, more random
(e)
larger, more random
ANOVA
93. What is the conclusion of the
hypothesis test? (p=0.0064, F test)
The data provide convincing evidence
that the average concentration
(a) is different for all groups.
(b) at level 1 is lower than the other
levels.
(c) is different for at least two of the
groups.
(d) is the same for all groups
ANOVA
94. The degrees of freedom for the F
test in a one-way ANOVA are:
(a)
(nT - r) and (r - 1)
(b)
(r - 1) and (nT - r)
(c)
(r – nT) and (nT - 1)
(d)
(nT - 1) and (r – nT)
ANOVA
95. After rejecting the null hypothesis
of equal treatments, a researcher
decided to compute a 95% confidence
interval for the difference between the
mean of treatment 1 and mean of
treatment 2 based on Tukey's
procedure. At α = 0.05, if the
confidence interval includes the value
of zero, then we can reject the
hypothesis that the two population
means are equal.
(a)
True
(b)
False
ANOVA
96. The Tukey procedure is used:
(a)
to test for normality.
(b) to test for homogeneity of
variance.
(c)
to test independence of errors.
(d) to test for differences in pairwise means
ANOVA
97. Suppose a variable has 3 levels:
high, middle, and low. If
α = 0.05 (FWER), what should be the
modified significance level (using
Bonferroni) for two sample t- tests for
determining which pairs of groups
have significantly different means?
(a) α* = 0.05
(b) α* = 0.05/2 = 0.025
(c) α* = 0.05/3 = 0.0167
(d) α* = 0.05/6 = 0.083
ANOVA
98. Based on the p-values given
below, which pairs of means are
significantly different? (Use the
Bonferroni criterion)
t-test between low & high: p=0.0052
t-test between low & mid: p=0.1236
t-test between mid & high: p=0.0544
(a) low & high
(b) mid & high
(c) low & mid; mid & high
(d) low & mid; low & high; mid & high
(e) none of the above
ANOVA
99. Which procedure cannot be used
for data snooping?
(a)
Bonferroni
(b)
Tukey
(c)
Scheffé
(d)
they can all be used
ANOVA
100. If the F-test for treatments is not
significant but the t-test for one of the
contrasts is significant, is it proper to
report the contrast?
(a) Yes
(b) Only if the contrast was planned
in advance.
(c) If you adjust for multiple
comparisons and it is still significant.
(d) Maybe
(e) No
ANOVA
101. In a balanced two-way ANOVA
the degrees of freedom for the error
term are (n is group size):
(a)
(a - 1)(b - 1)
(b)
ab(n - 1)
(c)
(a - 1)
(d)
abn + 1
ANOVA
102. In a balanced two-way ANOVA
the degrees of freedom for the
interaction term are:
(a)
(a - 1)(b - 1)
(b)
ab(n - 1)
(c)
(a - 1)
(d)
abn + 1
ANOVA
103. If you are comparing the
average sales between two different
brands you are dealing with a twoway ANOVA design.
(a)
True
(b)
False
ANOVA
104. In a two-way ANOVA, the
interpretations of the main effects (on
their own) make sense only when the
interaction component is not
significant.
(a)
True
(b)
False
ANOVA
105. In a two-way ANOVA setting,
when there are more than two levels
of a factor and there is no significant
interaction effect, Tukey's multiple
comparison procedure can be used to
perform pair-wise mean comparisons.
(a)
(b)
True
False
ANOVA
106. If a plot of the means of Factor B
verses the Factor A treatments
produces parallel plots, then one
could conclude that there was an
interaction effect between A and B.
(a)
True
(b)
False
ANOVA
107. In two-way ANOVA, what should
you always test first?
(a) The significance of factor A.
(b) The significance of factor B.
(c) The interaction between factors A
and B.
(d) None of the above.
ANOVA
108. Based on the results of a two-way
ANOVA (without interaction), the SSE
was computed to be 139.42. If we ignore
one of the factors and perform a one-way
ANOVA using the same data, will the SSE
be smaller than 139.42?
(a) Yes. Using the same data, the SSE
based on a one-way ANOVA is always
smaller than the SSE based on a two-way
ANOVA.
(b) No. Using the same data, the SSE
based on a one-way ANOVA is never
smaller than the SSE based on a two-way
ANOVA.
(c) Possibly. The SSE based on a one-way
ANOVA could be smaller or larger than
the SSE based on a two-way ANOVA.
ANOVA
109. The R2 from the interactive
model is 0.928 and the R2 from the
additive model is 0.893. All else
being equal (i.e. response is the
same!), we should take the model
with interaction because the R2 is
higher.
(a) TRUE
(b) FALSE
ANOVA
110. Why is balance important?
(a) easier calculations
(b) easier interpretations
(c) increased power
(d) some of the above
(e) all of the above
The simple row and column means in a
balanced table produce simple estimate of
model parameters.
To see that the treatment effects (A) are not
influenced by block differences (B), try adding
a certain fixed amount to all responses in one
block. It will not change any of the treatment
differences in a balanced table. But it will in
an unbalanced table.
ANOVA
111. If the interaction is significant
but the main effects aren’t:
(a) report on the significance of the
main effects
(b) remove the main effects from the
SSTR
(c) avoid talking about main effects
(d) test whether the main effects are
significant without interaction in the
model