PROBLEM SET #2
Multiple Choice Questions
1. Simple linear regression analysis determines
a) the true value of the population slope coefficient.
b) the linear relationship between two random variables.
c) the linear relationship between many different random variables.
d) the true value of the population intercept.
2.
In simple linear regression analysis, the dependent variable
a) is the variable that changes in response to changes in an independent variable.
b) is the variable whose changes affect the dependent variable.
c) is on the right-hand side variable.
d) can be either variable.
3.
A probabilistic relationship is one for which
a) the value of the dependent variable is perfectly determined by the value of the
independent variable.
b) all data points fall on the same line.
c) there is no random error component.
d) the value of the dependent variable is related to the value of the independent variable
but the data points do not fall on the same line.
4.
A marginal effect indicates
a) the predicted value of the dependent variable, holding all else constant.
b) the effect that a one-unit change in the independent variable is expected to have on the
dependent variable, holding all else constant.
c) the effect that a one-unit change in the dependent variable is expected to have on the
independent variable, holding all else constant.
d) the predicted value of the dependent variable when the independent variable equals ,
holding all else constant.
5.
The predicted value of is
a) the value that takes on when equals 0.
b) the effect that a one-unit change in the dependent variable is expected to have on the
independent variable, holding all else constant.
c) the value of when the slope is multiplied a specific and then that value is added to
the intercept.
d) the observed value of the dependent variable that is associated with a specific value of
the independent variable.
6.
The error term includes all of the following except
a) omitted variables.
b) deterministic relationships.
c) incorrect functional form.
d) measurement error.
7.
The estimated slope coefficient is
a) the estimated marginal effect of on .
b) the estimated value of when equals 0.
c) equal to the population slope coefficient.
d) .
8.
The estimated slope coefficient is
a) the ratio of the variance of to the variance of .
b) the ratio of the variance of to the variance of .
c) the ratio of the covariance between and to the variance of .
d) the covariance between and .
9.
The residual is
a) the vertical distance between the observed value of and the mean value of .
b) the vertical distance between the predicted value of and the mean value of .
c) the vertical distance between the observed value of and the probabilistic value of .
d) the vertical distance between the observed value of and the predicted value of .
10. We determine the estimated sample regression function by
a) minimizing the total deviations from the mean.
b) maximizing the sum of squared residuals.
c) minimizing the sum of squared residuals.
d) minimizing the least absolute deviations from the mean.
11. The explained variation in is
a) the distance between the best-fit line and the data points.
b) the distance between the mean and the predicted value of .
c) the distance between the mean and the data points.
d) the distance between the observed and predicted values of y.
12. The coefficient of determination
is
a) the ratio of the unexplained variation in to the total variation in .
b) the ratio of the explained variation in to the total variation in .
c) the ratio of the explained variation in to the unexplained variation in .
d) the ratio of the unexplained variation in to the explained variation in .
13. The standard error of the estimated sample regression function
is
a) the square root of the unexplained sum of squares.
b) the square root of the explained sum of squares.
c) the square root of the unexplained sum of squares divided by the degree of freedom of
the regression.
d) the square root of the explained sum of squares divided by the degree of freedom
14. When the estimated slope coefficient in the simple regression model, ̂ 1 , is zero,
a. R2 = Y .
b. 0 < R2 < 1.
c. R2 = 0.
d. R2 > (ESS/TSS).
Analytical questions: Remember your tools and how to develop tools from your tools.
1.
Consider the regression model : Yi = 0 + 1 Xi + i , where Yi denotes average test scores
from fifth-grade classes (TS, measured in percentages) and Xi denotes the data on fifth grade
class size (CS) . You collected data on 25 classes with class sizes ranging from 25 to 36
and obtained the following measures
X = 75 Y = 50, X2 = 625 Y2 = 228, XY = 30
a. Find the estimated regression equation and interpret it fully.
b. Find the R2 and interpret the value. What are the units of measurement for the R 2?
d. A classroom has 22 students. What is the regression prediction for the classroom’s
average test score? Is this prediction reliable? Why or why not?
c. The sample average class size across the 25 classrooms is 5. What is the average test
scores across the 25 classrooms?
2. (in this exercise, you’d be manipulating some of the formulas/concepts discussed to arrive
at your solutions)
In a simple regression and correlation analysis based on 72 observations, we find r = 0.8
and SY|X=10
(a)
Find the amount of unexplained variation (USS or Residual Sum of Squares)
(b)
Find the proportion of unexplained variation to the total variation.
(c)
Find the total variation of the dependent variable (TSS)
3
A counselor working with teenagers is interested in the relationship between anxiety
and depression. The counselor administers a depression and anxiety test to each teenagers
selected randomly. The scores obtained from the administration of the two inventories are
summarized below.
The summary statistics are
Anxiety
Depression
Sample size
19
19
Minimum
0
2
Maximum
68
33
Sample mean
Standard deviation
Sample co-variation x,y
3828.0526
a. If anxiety is the independent variable and depression is the dependent
variable, what is the sample regression function? Interpret your estimated
regression equation. Find the coefficient of determination and interpret it
3. A company sets prices for Heineken beer (an imported beer) in 200 different areas.
Consider the regression model : Yi = 0 + 1 Xi + i , where Yi denotes the monthly
sales (SALES, measured in thousands of dollars) and Xi denotes the Price (measured
in dollars) The range of the price values used in the samples is from $1.35 to $28.50.
a. What does the error term I represent? Be specific in your answer (Refer to
question 4.4 in your text for some ideas)
Suppose the regression above yields the following:
b. Interpret the regression equation fully.
c
Interpret the R2. What are the units of measurement for the R2?