Test questions for Lessons 5-7 – Simple regression
1. The following simple regression equation is given: y’ = 5 - 0.78X. What can we know
from this equation:
a) The correlation between X and Y is positive
b) The correlation between X and Y is (-0.78)
c) For every 1 point increase in X, the predicted y value decreases by 0.78
d) Answers (b) and (c) are correct
Explanation:
The given formula is an unstandardized simple regression formula (we predict y’ and not
Zy’). In the given formula, we can see that b is negative which means that the correlation
in negative. However, b does not equal r (only in standardized simple regression equation
β=r). The meaning of b is the change in y’ for every one unit increase in X. Thus, only
answer c is correct.
2. Which of the following produces a better prediction in a simple regression?
a) Larger sum of squared differences between actual y values and the mean of y
b) Larger sum of squared differences between predicted y values and the mean of y
c) Smaller sum of squared differences between actual y values and predicted y
values
d) Answers (b) and (c) are correct
e) All of the above (a, b and c are correct)
Explanation:
A prediction in simple regression is better when the proportion of explained variance is
large, and the proportion of the unexplained variance is low.
The squared differences between actual y values and the mean of y = SST. A large SST
does not say anything about the prediction (only about the variance in y).
The sum of squared differences between predicted y values and the mean of y =
SSregression. A large SSregression means that we explained a large proportion of
variance, which denotes a better prediction.
The sum of squared differences between actual y values and predicted y = SSE. A small
SSE means that the unexplained variance is small, which denotes a better prediction.
3. In an unstandardized simple regression model in which r is positive, for every 1 point
increase in X, the ____ goes up by ___ units.
a) actual y; b
b) actual y; β
c) predicted y; b
d) predicted y; β
Explanation:
, The regression equation gives the predicted y based on a value of X. In an unstandardized
regression the coefficient is b and not beta. The definition of b is as follows: For every
one point increase in X, y goes up by b units. In a standardized regression the coefficient
is beta. The definition of beta is as follows: For every one point increase in Zx, Zy’ goes
up by beta units. In other words, for every one SD increase in X, Zy’ goes up by beta
SDs.
4. Which of the following scenarios reflects a regression model with the best predictive
value?
a) SEest = 1
𝑆𝑆𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛
b) =1
𝑆𝑆𝑡𝑜𝑡𝑎𝑙
𝑆𝑆𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠
c) =1
𝑆𝑆𝑡𝑜𝑡𝑎𝑙
𝑀𝑆𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛
d) =1
𝑀𝑆𝑟𝑒𝑠𝑖𝑑𝑢𝑙𝑠
Explanation:
A prediction in simple regression is better when the proportion of explained variance out
of total variance is large. If SSregression \ SStotal = 1 it means that all of the variance is
explained. If SSresiduals \ SStotal = 1 it means that all of the variance is unexplained,
which depicts the worst prediction case. MSregression \ MSresiduals equals F. A large F
would mean that the effect is likely to be significant, but if F=1 it means that the effect is
not significant. SEest = 1 means that the average error in prediction is 1, this does not
mean that the prediction is strong or weak, as it depends on the scaling.
5. Two researchers conducted studies in order to predict Y from X. They both sampled the
same amount of people and calculated the same simple regression line. Also: in both
studies, SST was the same. However, the first researcher calculated a smaller SEest. This
means that the first researcher:
a) Found a stronger correlation between x and y
b) For any given x, found a smaller 95% confidence interval for y
c) Had a higher critical F value
d) Answers (a) and (b) are correct
e) All of the above (a, b and c are correct)
Explanation:
SEest is the average error in prediction. A small average error means that the prediction is
better. That is, the correlation between X and Y is stronger. Also, as the confidence
interval is based on SEest, a small SEest would mean a smaller confidence interval. SEest
does not affect the critical F value. The critical F value is determined by the degrees of
freedom (k,n-k-1), which are just the same for both researchers.
6. A researcher conducted a study to predict the amount of sushi a person consumes per
month from his/her swimming ability. He found the following regression equation: Z'y =
1. The following simple regression equation is given: y’ = 5 - 0.78X. What can we know
from this equation:
a) The correlation between X and Y is positive
b) The correlation between X and Y is (-0.78)
c) For every 1 point increase in X, the predicted y value decreases by 0.78
d) Answers (b) and (c) are correct
Explanation:
The given formula is an unstandardized simple regression formula (we predict y’ and not
Zy’). In the given formula, we can see that b is negative which means that the correlation
in negative. However, b does not equal r (only in standardized simple regression equation
β=r). The meaning of b is the change in y’ for every one unit increase in X. Thus, only
answer c is correct.
2. Which of the following produces a better prediction in a simple regression?
a) Larger sum of squared differences between actual y values and the mean of y
b) Larger sum of squared differences between predicted y values and the mean of y
c) Smaller sum of squared differences between actual y values and predicted y
values
d) Answers (b) and (c) are correct
e) All of the above (a, b and c are correct)
Explanation:
A prediction in simple regression is better when the proportion of explained variance is
large, and the proportion of the unexplained variance is low.
The squared differences between actual y values and the mean of y = SST. A large SST
does not say anything about the prediction (only about the variance in y).
The sum of squared differences between predicted y values and the mean of y =
SSregression. A large SSregression means that we explained a large proportion of
variance, which denotes a better prediction.
The sum of squared differences between actual y values and predicted y = SSE. A small
SSE means that the unexplained variance is small, which denotes a better prediction.
3. In an unstandardized simple regression model in which r is positive, for every 1 point
increase in X, the ____ goes up by ___ units.
a) actual y; b
b) actual y; β
c) predicted y; b
d) predicted y; β
Explanation:
, The regression equation gives the predicted y based on a value of X. In an unstandardized
regression the coefficient is b and not beta. The definition of b is as follows: For every
one point increase in X, y goes up by b units. In a standardized regression the coefficient
is beta. The definition of beta is as follows: For every one point increase in Zx, Zy’ goes
up by beta units. In other words, for every one SD increase in X, Zy’ goes up by beta
SDs.
4. Which of the following scenarios reflects a regression model with the best predictive
value?
a) SEest = 1
𝑆𝑆𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛
b) =1
𝑆𝑆𝑡𝑜𝑡𝑎𝑙
𝑆𝑆𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠
c) =1
𝑆𝑆𝑡𝑜𝑡𝑎𝑙
𝑀𝑆𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛
d) =1
𝑀𝑆𝑟𝑒𝑠𝑖𝑑𝑢𝑙𝑠
Explanation:
A prediction in simple regression is better when the proportion of explained variance out
of total variance is large. If SSregression \ SStotal = 1 it means that all of the variance is
explained. If SSresiduals \ SStotal = 1 it means that all of the variance is unexplained,
which depicts the worst prediction case. MSregression \ MSresiduals equals F. A large F
would mean that the effect is likely to be significant, but if F=1 it means that the effect is
not significant. SEest = 1 means that the average error in prediction is 1, this does not
mean that the prediction is strong or weak, as it depends on the scaling.
5. Two researchers conducted studies in order to predict Y from X. They both sampled the
same amount of people and calculated the same simple regression line. Also: in both
studies, SST was the same. However, the first researcher calculated a smaller SEest. This
means that the first researcher:
a) Found a stronger correlation between x and y
b) For any given x, found a smaller 95% confidence interval for y
c) Had a higher critical F value
d) Answers (a) and (b) are correct
e) All of the above (a, b and c are correct)
Explanation:
SEest is the average error in prediction. A small average error means that the prediction is
better. That is, the correlation between X and Y is stronger. Also, as the confidence
interval is based on SEest, a small SEest would mean a smaller confidence interval. SEest
does not affect the critical F value. The critical F value is determined by the degrees of
freedom (k,n-k-1), which are just the same for both researchers.
6. A researcher conducted a study to predict the amount of sushi a person consumes per
month from his/her swimming ability. He found the following regression equation: Z'y =
