ECON 424/524 Homework 1: Solutions to Exercises Spring 2005
2.1 (a) (b)
(c) (d)
(e) (f)
2.3 (a)
(b) The probability that, on a given Monday, either 2, or 3, or 4 students will be absent is
.
(c) The probability that, on a given Monday, more than 3 students are absent is
(d)
3.066 is the average number of students absent on Mondays after considering infinitely many Mondays.
(e)
= 10.5776
(f)
2.11 var(a+cX) = E[(a+cX) - E(a+cX)]2 = E[a+cX - a - cE(X)]2
= E[c(X - E(X))]2 = c2E[X - E(X)]2 = c2var(X)
2.21
Thus, Z ~ N(0,1).
3.12 (a) If the simple linear regression model becomes
(b) Graphically, setting implies the mean of the simple linear regression model passes through the origin (0, 0).
(c) The least-squares estimators in (3.3.8) are no longer appropriate if we know because, in this case, a different least squares “sum of squares” function needs to be minimized.
(d) To save on subscript notation we set The sum of squares function becomes
A plot of this quadratic function appears in Figure 3.10. The minimum of this function is approximately 12 and occurs at approximately The significance of this value is that it is the least-squares estimate.
2.3 (e) To find the value of b that minimizes we obtain
Setting this derivative equal to zero, we have
or
Thus, the least-squares estimate is
which agrees with the approximate value of 1.95 that we obtained geometrically.
(f) The computer generated value is also
(g) The fitted regression line is plotted in Figure 3.11. Note that the point does not lie on the fitted line in this instance.
(h) The least squares residuals, obtained from are:
Their sum is Note this value is not equal to zero as it was for
Figure 3.10 Sum of squares for Figure 3.11 Fitted regression: .
3.14 (a) The consultant’s report implies that the least squares estimates satisfy the following two equations
Solving these two equations yields
(b) A graph of the estimated regression line appears in Figure 3.13.
Figure 3.13 Graph of sales-advertising regression line for Exercise 3.14
Homework 1- Empirical Exercises
Question 1(a)
Obs Q F
1 1250 81
2 890 60
3 1093 73
4 1546 86
5 635 58
6 937 68
7 1142 75
8 1120 76
9 1067 72
10 1410 84
11 987 69
12 1198 77
13 1429 85
14 1147 74
15 1200 74
16 904 62
17 1342 83
18 1005 70
Question 2(a)
Part 2(b): Increases.
Question 3(a):
The MEANS Procedure
Variable Mean Median Range Minimum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
q 1127.89 1131.00 911.0000000 635.0000000
f 73.7222222 74.0000000 28.0000000 58.0000000
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Variable Maximum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
q 1546.00
f 86.0000000
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
The mean of q is 1127.89, the median is 1131, and the range is 911. The mean of f is 73.72, the median is 74 and the range is 28.
Part (b): It is difficult to know if the soda sales figures are reasonable, but the temperature figures, with measures of central tendency near 74 degrees Fahrenheit and a range of 28 degrees, seem reasonable.
Question 4(a)
The REG Procedure
Model: MODEL1
Dependent Variable: Q
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 786802 786802 225.86 <.0001
Error 16 55738 3483.62591
Corrected Total 17 842540
Root MSE 59.02225 R-Square 0.9338
Dependent Mean 1127.88889 Adj R-Sq 0.9297
Coeff Var 5.23298
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 -771.26409 127.13317 -6.07 <.0001
F 1 25.76093 1.71413 15.03 <.0001
Part (b):
b2 = 25.76 à If the temperature increases by 1 degree, soda sales are expected to increase by 25.8 units (we don’t know if it is measured in cans, cups, cases, etc, so we have to just say units).
Part (c): According to economic theory, in general the quantity demanded depends on price, prices of substitutes and complements, tastes, income, and population (in this case, tickets sold). Including these variables would improve the model.
Part (d):
Part (e): A point on the regression line represents the value of y predicted from the regression estimates for a given value of x. That is, the point is (x0,y0) where y0 = b1 + b2x0.
Log File
28 options pageno=1 nocenter ps=45 ls=70;
29 *ECON 424/524 Homework 1;
30
NOTE: PROCEDURE REG used:
real time 1:27.73
cpu time 0.31 seconds
31 data pigskin;
32 infile
33 'C:\Documents and Settings\Carol
33 ! Tremblay\Desktop\424_524\football_dat.txt';
34 input q f;
NOTE: The infile 'C:\Documents and Settings\Carol
Tremblay\Desktop\424_524\football_dat.txt' is:
File Name=C:\Documents and Settings\Carol
Tremblay\Desktop\424_524\football_dat.txt,
RECFM=V,LRECL=256
NOTE: 18 records were read from the infile 'C:\Documents and
Settings\Carol Tremblay\Desktop\424_524\football_dat.txt'.
The minimum record length was 6.
The maximum record length was 7.
NOTE: The data set WORK.PIGSKIN has 18 observations and 2 variables.
NOTE: DATA statement used:
real time 0.03 seconds
cpu time 0.03 seconds
38 proc print;
39 run;
NOTE: There were 18 observations read from the data set WORK.PIGSKIN.
NOTE: PROCEDURE PRINT used:
real time 0.01 seconds
cpu time 0.01 seconds
40
42 proc plot;
43 plot q*f;
44 run;
45
NOTE: There were 18 observations read from the data set WORK.PIGSKIN.
NOTE: PROCEDURE PLOT used:
real time 0.01 seconds
cpu time 0.01 seconds
47 proc means mean median range min max;
48 run;
NOTE: There were 18 observations read from the data set WORK.PIGSKIN.
NOTE: PROCEDURE MEANS used:
real time 0.00 seconds
cpu time 0.00 seconds
49
51 proc reg;
52 model q = f;
53 plot q*f;
54 run;
NOTE: 18 observations read.
NOTE: 18 observations used in computations.
NOTE: Graph's name, REG, changed to REG1. REG is already used or not
a valid SAS name.