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QBUS2810 Statistical Modelling for Business
Q1 Why in OLS for SLR is the sample average error, eˉ = 1n
∑n
i=1 ei = 0?
(a) Because it is an error term , it has to average 0.
(b) Because each error ei is equal to 0, therefore the average of a set of 0s is 0.
(c) Because when we do OLS, we take the 1st derivative of the RSS, and the derivative
with respect to β1 is ?2× the sum of the errors times X, i.e. ?2 ×
∑n
i=1 eiXi. We
set this sum equal to 0 to get the LS estimate. Thus eˉ = 0
(d) Because when we do OLS, we take the 1st derivative of the RSS, and the
derivative with respect to β0 is ?2× the sum of the errors, i.e. ?2×
∑n
i=1 ei.
We set this sum equal to 0 to get the OLS estimate. Thus eˉ = 0.
(e) None of the above are correct.
Q2 True or False? LSA 2 states that E(εi|Xi) = 0. This implies that the residual series ε
and X are uncorrelated. Answer: True
Q3 LSA 2 states that E(εi|Xi) = 0. This implies that the error series ε and X are uncorre-
lated, because:
(a) Other factors always exist and are implicitly affecting Y through ε, thus ε and X
must be uncorrelated.
(b) ε is an i.i.d error series and hence must be uncorrelated with X.
(c) If they were correlated, then the slope of the regression of εi on Xi would
not be 0, i.e. we could write E(εi|Xi) = γ0 + γ1Xi and γ1 6= 0. Thus LSA 2
would not be correct.
(d) Other factors always exist and are implicitly affecting Y through ε, thus ε and X
must be correlated. Hence, LSA 2 does not imply they are uncorrelated.
Why does RSS always decrease when you add another X variable to the regression model?
Q4
(a) Because the new X variable is always significantly related to Y
2(b) Because now there is one extra parameter with which to optimize RSS, meaning a
more optimum, hence lower, RSS can be found.
(c) Because the OLS estimate of the new X’s regression slope will not be exactly 0.
(d) It doesn’t, sometimes RSS increases or stays the same, e.g. if the new X variable is
not related to Y.
(e) Both (b) and (c) are true.
Q5 Would the variable number of children cause OVB regarding the effect of Salary on
Amount Spent?
(a) Number of children would not be correlated with Salary, so: NO.
(b) Number of children is likely correlated with Salary, but it would not be a factor
determining Amount Spent, so: NO.
(c) Number of children is likely correlated with Salary. Also, number of children could
be a factor determining Amount Spent, so: YES.
(d) Even though number of children is a likely determinant of Amount Spent, it would
not be correlated with Salary, so NO.
(e) We should first look at the sample correlation between number of children
and Salary here. Then, decide whether number of children could be a
determinant of Amount Spent.
Q6 Would the variable IQ level cause OVB regarding the effect of Salary on Amount Spent?
(a) It is likely that IQ is correlated with Salary. It is unlikely that IQ is a
determinant of Amount Spent for a company like Direct Marketing which
sells clothing,books and sports gear. So: NO.
(b) IQ would not be correlated with Salary nor would it determine Amount Spent, so
NO.
(c) IQ would be correlated with Salary and thus also be correlated with Amount Spent,
since Salary is correlated with Amount Spent. Thus, YES.
(d) IQ would not be correlated with Salary, but it would help determine Amount Spent,
so NO.
Q7 The Mann-Whitney U test is preferred to the t-test whenever:
(a) The dataset in each group has a large enough sample size, ni, for the central limit to
work, i.e. ni ≥ 30.
(b) The data has no outliers and has a symmetric shaped distribution in each group.
3(c) The data has some outliers and it is unclear if E(Y 4) <∞ in each group.
(d) The data are on the ordinal scale.
(e) Both (c) and (d) are correct.
Q8 The t-test for a mean (difference) is very popular and mostly used in practice because:
(a) It has comparatively high power and is also robust to outliers.
(b) Its properties are very well known under the LSA; e.g. BLUE, consistency,
etc
(c) It has higher power than both the Mann-Whitney and median tests, for data with
infinite 4th moments.
(d) It has lower power than both the Mann-Whitney and median tests, for data with
infinite 4th moments.
(e) None of the above.