2. Decisionmaking is a process by
which "best solutions‖ are
found.
Managers are the one who make
the decisions to lead to the best
outcome possible under a given
circumstances.

3. Itis an act, process or
methodology of making
something fully perfect,
functional or as effective as
possible.

4. Case of a MANAGER
- A manager always find the level
of output that maximizes the
profit of the firm or to determine
how much labor, capital and raw
material inputs to use to produce
a given amount of output at the
lowest possible cost.

5. Case of Consumers
- As consumers, they will search goods
within the constraints imposed by
their prices and their income, for the
combination of goods and services that
will yield the highest level of
satisfaction.

6. The function the decision maker
seeks to maximize or minimize
Examples:
1. Manager – will always try to
maximize profit.
2. Consumer – will always
maximize consumer goods.

7. - Optimization problem that involves
maximizing/minimizing the objective
function.

8. - When the Objective function
measures a benefit, the decision
maker seeks to maximize this benefit
thus solving a maximization problem.
- When the Objective function
measures a cost, the decision maker
seeks to minimize the cost, thus
solving a minimizing problem.

9.  Determines the objective function.
Example:
- Profit
The value of profit will be determined by the
number of units sold or produced while the
production of unit of the good is the activity or
choice variable that determines the value of the
objective function which is profit.

11.  ObjectiveFunction- Measures whatever it is
that the particular decision maker wishes to
either maximize or minimize.
 E.g. profit, cost, satisfaction…
 Maximization Problem- optimizing problem
that involves maximizing the objective
function

12.  DiscreteChoice Variable- choice Variable
that can only take a specific integer
 ContiniousChoice Variable- choice variable
that can take on any value between two end
point.

13.  Minimization Problem- optimizing problem
that involves minimizing the objective
function
 Activities
or Choice variables- Determine the
value of Objective function.
 Objectivefunction maybe a function of more
than one activity

14.  Unconstrained optimization- an optimization
problem wherein the decision maker can
choose any level of activity from unrestricted
set of values.
 E.g.no external restrictions inchoosing any
level of output in order to maximize net
benefit.

15.  Constrained Optimization- Optimization
problems wherein the decision maker can
choose values for choice variables from a
restricted set of values
 Constrained maximization- maximization
problem where activities must be chose to
satisfy a side constraint that the total cost of
activities be held to specific amount
 Total benefit function=objective function
 Total cost= constraint

16.  Constrained Minimization- minimization
problem where the activities must be chosen
to satisfy a side constraint that the total
benefit of the activities be held to specific
amount.
 Objective function= Total cost function
 Total benefit function= constraint
 E.g. gift shop

17.  MarginalAnalysis – analytical tool for solving
optimization problems that involves changing
the value of choice variable by a small
amount to see if the objective function can
be further increased or further decreased

18.  Unconstrained Maximization
 NB= TB-TC
 NB=net benefit
 TB=total benefit
 TC=total cost
 The activity is increased or decreased in
order to obtain highest level of benefit
 The optimal level of activity is obtained
when no further increase in net benefit are
possible in any change of activity

19.  MarginalBenefit- addition to total benefit
attribute to increasing the activity by a small
amount.
 MarginalCost- addition to total cost attribute
to increasing the activity by a small amount

20. MB= Change in total benefit
Change in activity
MC= Change in total cost
Change in activity
MB>MC MB<MC
Increase activity NB rises NB falls
Decrease activity NB falls NB rises
Optimal level of the activity is attained-net benefit is
maximized-when level of activity is the last level for which
marginal benefit exceeds marginal cost

21.  Maximization with a Continuous Choice variable
When a decision maker wishes to obtain the
maximum net benefit from an activity that is
continuously variable, the optimal level of the
activity is that level at which the marginal
benefit is equal to marginal cost (MB=MC)

22. Constrained Optimization
- An objective function is maximized or minimized
subject to a constraint if, for all of the activities
in the objective function, the ratios of marginal
benefit per dollar spent be equal for all
activites.
- MBA/PA=MBB/PB

24.  Thischapter set forth the basic principles of
regression analysis: estimation and
assessment of statistical significance. We
emphasized how to interpret the results of
regression analysis, rather than focusing on
the mathematics of regression analysis.

25.  The coefficients in an equation that
determine the exact mathematical relation
among variables.
 Y being the dependent variable and X the
independent or the explanatory variable.

26.  Itis the process of finding estimates of the
numerical values of the parameters of
equation.

27.  Thetwo variable linear model or the simple
regression analyisis is used for testing
hypothesis using the Y variable or the
independent variable and X variable or the
explanatory varible.

28. YEAR n Yi(Corn) Xi(fertilizer) yi xi xiyi xi2
1971 1 40 6 -17 -12 204 144
1972 2 44 10 -13 -8 104 64
1973 3 46 12 -11 -6 66 36
1974 4 48 14 -9 -4 36 16
1975 5 52 16 -5 -2 10 4
1976 6 58 18 1 0 0 0
1977 7 60 22 3 4 12 16
1978 8 68 24 11 6 66 36
1979 9 74 26 17 8 136 64
1980 10 80 32 23 14 322 196
Total: 10 570 180 0 0 956 576
mean: 57 18

29. YEAR n Yi(Corn) Xi(fertilizer)
1971 1 40 6
1972 2 44 10
1973 3 46 12
1974 4 48 14
1975 5 52 16
1976 6 58 18
1977 7 60 22
1978 8 68 24
1979 9 74 26
1980 10 80 32

30. YEAR n Yi(Corn) Xi(fertilizer)
1971 1 40 6
1972 2 44 10
1973 3 46 12
1974 4 48 14
1975 5 52 16
1976 6 58 18
1977 7 60 22
1978 8 68 24
1979 9 74 26
1980 10 80 32
Total: 10 570 180
mean: 57 18

31. YEAR n Yi(Corn) Xi(fertilizer) yi
1971 1 40 6 -17
1972 2 44 10
1973 3 46 12
1974 4 48 14
1975 5 52 16
1976 6 58 18
1977 7 60 22
1978 8 68 24
1979 9 74 26
1980 10 80 32
Total: 10 570 180
mean: 57 18

32. YEAR n Yi(Corn) Xi(fertilizer) xi
1971 1 40 6 -12
1972 2 44 10
1973 3 46 12
1974 4 48 14
1975 5 52 16
1976 6 58 18
1977 7 60 22
1978 8 68 24
1979 9 74 26
1980 10 80 32
Total: 10 570 180
mean: 57 18

33. YEAR n Yi(Corn) Xi(fertilizer) yi xi xiyi
1971 1 40 6 -17 -12 204
1972 2 44 10 -13 -8
1973 3 46 12 -11 -6
1974 4 48 14 -9 -4
1975 5 52 16 -5 -2
1976 6 58 18 1 0
1977 7 60 22 3 4
1978 8 68 24 11 6
1979 9 74 26 17 8
1980 10 80 32 23 14
Total: 10 570 180 0 0
mean: 57 18

34. YEAR n Yi(Corn) Xi(fertilizer) yi xi xi2
1971 1 40 6 -17 -12 144
1972 2 44 10 -13 -8 64
1973 3 46 12 -11 -6 36
1974 4 48 14 -9 -4 16
1975 5 52 16 -5 -2 4
1976 6 58 18 1 0 0
1977 7 60 22 3 4 16
1978 8 68 24 11 6 36
1979 9 74 26 17 8 64
1980 10 80 32 23 14 196
Total: 10 570 180 0 0 576
mean: 57 18 xi2

35. b1 (slope of the estimated = XiYi / Xi2
regression line) = 1.66
b0 (Y intercept) = 27.13 = mean of Yi - (bi * mean of
Ŷi (estimated Regression Xi )
equation) = 27.12 + 1.66Xi

36. 90
80
70
60
50
Yi
40 Xi
30
20
10
0
1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981

37. Coeffi Standar t Stat P- Lowe Upper Lower Upper
cients d Error value r 95% 95% 95.0% 95.0%
Intercept 27.13 1.9792653 13.70457 7.74557E 22.5608 31.68919 22.56080 31.689194
48 984 -07 0593 407 593 07
X 1.66 0.1013210 16.38081 1.94353E 1.42607 1.893369 1.426075 1.8933690
87 745 -07 5378 067 378 67
Variable
1

38. Regression Statistics
Multiple R 0.985418303
R Square 0.971049232
Adjusted R Square 0.967430386
Standard Error 2.431706077
Observations 10

39. ANOVA
d SS MS F Significance F
f
Regression 1 1586.694444 1586.694444 268.3311803 1.94353E-07
Residual 8 47.30555556 5.913194444
Total 9 1634

44.  Tcomp is greater than the Tcrit (16.38081745>1.860).
Since that is the case then X variable is significant
with the margin of error given, which is 5%, to
explicate the relationship between X and Y
parameters.
• R2 or the explanatory power of the model is equal
to 0.9710 or 97.10%. This explains that fertilizer (X)
expresses 97.10% of output change in Corn . The R2 is
significantly different from zero.
• In F distribution, the Fcomp explains that the
parameters are not all equal to zero. The high value
of F ratio implies a significant relationship between
the dependent and independent variables.

45.  The test of significance of parameter
estimates passed as well as the test for the
coefficient of multiple determination and
test of the overall significance of the
regression.

46. Population Regression Line Sample Regression Line
The equation or line The line that best fits
representing the the data in the
true or (actual) sample is call the
relation between sample regression line
dependent variable
and the explanatory
Variable

47.  Anestimator that produces estimates of a
parameter that are on average equal to the
true value of the parameter

48.  Thedistribution (and relative frequency) of
values b can take because observations on Y
and X come from a random sample

49.  The estimated coefficient is far enough away
from zero
 Either sufficiently greater than zero (a
positive estimate) or sufficiently less than
zero (a negative estimate)

50.  t-stat is used to test the hypothesis that the
true value of b equals zero
 If the t-stat is greater than the critical value
of t, then the hypothesis that b=0 is rejected
in favor of the alternative hypothesis that
b not =0
 When the calculated t-stat exceeds the
critical value of t, b is significantly different
from zero, or equivalently, b is statistically
significant

51.  Using P-value
 We use P-value for analyzing data and to make the strongest possible
conclusion from the limited data’s that are given.
 To get the p-value, you need to have the estimated value then the
significance level of the alternative hypothesis then the test statistics.
Decision Criterion for a Hypothesis Test Using the P-value:
If P-value is less than a, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
Examples:
Ha: µ 30 versus Ho: µ = 30
Assumptions: X is normally distributed with s = 8
Test Statistic:
a = .05 RR: z < -1.96 or z >1.96 (P-value < .05)
Calculation: z = 1.54
P-value = 2P(z > |zcalculated|) = 2P(z > |1.54|) = 2P(z < -1.54)
= 2(.0618) = .1236
Decision: Fail to reject Ho.

52.  Evaluation of Regression Equation
 Regression Equation(y) = a + bx
Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
Intercept(a) = (ΣY - b(ΣX)) / N
where
x and y are the variables.
b = The slope of the regression line
a = The intercept point of the regression line and the y axis.
N = Number of values or elements
X = First Score
Y = Second Score
ΣXY = Sum of the product of first and Second Scores
ΣX = Sum of First Scores
ΣY = Sum of Second Scores
ΣX2 = Sum of square First Scores
Regression Example: To find the Simple/Linear Regression of
X Values 60 61 66 63 65
Y Values 3.1 3.2 3.8 4 4.1
To find regression equation, we will first find slope, intercept and use it to form regression
equation..
Step 1: Count the number of values.
N=5
Step 2: Find XY, X2

53. Step 3: Find ΣX, ΣY, ΣXY, ΣX2.
ΣX = 311
ΣY = 18.6
ΣXY = 1159.7
ΣX2 = 19359
Step 4: Substitute in the above slope formula given.
Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
= ((5)*(1159.7)-(311)*(18.6))/((5)*(19359)-(311)2)
= (5798.5 - 5784.6)/(96795 - 96721)
= 13.9/74
= 0.19
Step 5: Now, again substitute in the above intercept formula given.
Intercept(a) = (ΣY - b(ΣX)) / N
= (18.6 - 0.19(311))/5
= (18.6 - 59.09)/5
= -40.49/5
= -8.098
Step 6: Then substitute these values in regression equation formula
Regression Equation(y) = a + bx
= -8.098 + 0.19x.
Suppose if we want to know the approximate y value for the variable x = 64. Then we can
substitute the value in the above equation.
Regression Equation(y) = a + bx
= -8.098 + 0.19(64).
= -8.098 + 12.16
= 4.06

54.  Coefficient of determination
 It is used for statistical models whose main purpose is to predict the outcome of the
future based by other related information.
 Measures percentage variation in Y that can be explained by the X’s through the model
Y=Xβ + ε
 Proportionate reduction of total variation in Y associated with the use of the set of
independent variables X1, X2, …, Xk (assuming a constant term is included in the
model)
 A goodness of fit measure
Consider Tampa sales example. From printout, R2 = 0.9453.
• Interpretation: 94% of the variability observed in sale prices can be
explained by assessed values of homes. Thus, the assessed value of the
home contributes a lot of information about the home’s sale price.
• We can also find the pieces we need to compute R2 by hand in either
JMP or SAS outputs:
– SSyy is called Sum of Squares of Model in SAS and JMP
SSE is called Sum of Squares of Error in both SAS and JMP.
• In Tampa sales example, SSyy = 1673142, SSE = 96746 and thus
R2 = 1673142 − 96746/1673142 = 0.94.

55.  F-test
 F-test is a simultaneous test that if all of the beta=0 (it means that all of
your x’s are useless) or at least one x is not equal to zero (which means
that specific variable is affecting Y).
 Ex:. The hypothesis that the means of several normally distributed
populations, all having the same standard deviation, are equal. This is
perhaps the best-known F-test, and plays an important role in the
analysis of variance (ANOVA).
 The hypothesis that a data set in a regression analysis follows the
simpler of two proposed linear models that are nested within each other.

56.  Multiple Regression
 It’s purpose is to learn more about the relationship between
several independent and dependent variables.
 Ex. A car agent having a listing of the following cars in the
following characteristics of that car—transportation, comfort,
style, luxury, fuel economy, etc. Once these following
information has been compiled for the various cars, it will be
interesting to see whether and how these measures relate to
the price for which a car is sold. For example, the space of this
car is better analyst of the price of which a car sells than how
luxurious the car is.

57. Quadratic Regression Models
Log-Linear Regression Models

58. Are used when the underlying relation between
X and Y plots as a curve, rather than a
straight line.

61.  An analyst would use nonlinear regression
model when the scatter diagram shows a
curvilinear pattern.
 Nonlinear regression is a general technique
to fit a curve through your data.
 The purpose of linear regression is to find the
line that comes closest to your data.

62.  Quadratic Regression Model
 Log-Linear Regression Model

63.  One of the most useful nonlinear forms for
managerial economics
 expressed as Y = a + bX + cX2
 Nonlinear model to Linear model
 Create a new variable. ―Z‖ defined as Z = X2
 Y= a + bX + cZ

64.  Run Regression of Y from X and Z
Dependent Variable: Y F-Ratio: 13.11
R-
Y X Z Observations: 12 squared: 0.75
82 3 9
107 3 9 Parameter Standard T-
61 4 16 Variable Intercept Error Ratio
77 5 25
68 6 36 Intercept 140 17.14 8.17
30 8 64 X -20 4.14 -4.83
57 10 100 Z 1.01 0.5 2
40 12 144
82 14 196
68 15 225
102 17 289
110 18 324

65.  Estimated quadratic regression equation is
 Y = 140.08 – 19.51X + 1.01X2
 1.01 is also the slope parameter estimate for
X2
 The estimated equation can be used to
estimate the value of Y for any particular
value of X

66.  Example: if X = 10
 Y will be equal to 45.98
Y=140.08 – 19.51(10) + 1.01(10)2
 After which, perform a t-tests to determine
the statistical significance of each
parameters.

67. Y is related to one or more explanatory variables
in a multiplicative form
 Y=aXb Zc
 Transform to a linear equation
Percentage change in Y
• b 
Percentage change in X
Percentage change in Y
• c 
Percentage change in Z

68.  Parameters b and c are elasticities.
 To transform the equation in to a
linear form, we must use the natural
logarithms of both sides of the
equation.
 lnY = (ln a) + b(ln X) + c(ln Z)
 If we define: Y’ = a’ + bX’ + cZ’

69.  Example
Variable: Y = aXb
 Since Y is positive at all points, parameter a is expected to be positive.
 Since Y is decreasing as X increases, the parameter X(b) is expected to be negative.
Y X
2810 20
2825 30
2031 30
2228 40
1620 40
1836 50
1217 60
1110 90
1000 110
420 120
602 140
331 170

70.  To estimate the parameters a and b in a nonlinear
equation, we transform the equation by taking
logarithms:
lnY = ln a + b lnX
LOG Y LOG X
7.94094 2.99573
7.94626 3.4012
7.61628 3.4012
7.70886 3.68888
7.39018 3.68888
7.51534 3.91202
7.10414 4.09434
7.01212 4.49981
6.90776 4.70048
6.04025 4.78749
6.40026 4.94164
5.80212 5.1358

71.  Run Regression
Dependent
Variable LOG Y F-Ratio 70
Observations 12 R-Square 0.875
Parameter Standard T-
Variable Intercept Error Ratio
Intercept 11.06 0.48 23.04
Log X -0.96 0.11 -8.73

72.  To obtain parameter estimates:
note: the slope parameter on Log X is also the exponent on X in the
linear equation.
Y=aX(-0.96)
note: since b is an elasticity, 10 percent increase in X results in a 9.6
percent decrease Y.
 To obtain the estimate of a:
note: we take the antilog of the estimated value of the intercept
parameter.
= antilog(11.06)
= 63, 576
Y=63,576X(-0.96)

73.  Showthat the two models are
mathematically equivalent.
logX = 4.5
logY = 6.74 = [11.06 – 0.96(4.5)]
take the antilog of Y and X.
X = 90 Y = 845
Y = 845 = [63,577(90) (-0.96) ]

74.  Regression analysis is simply a tool to provide
the necessary information for a manager to
make decisions that maximizes profits.
 Itoffers managers a way of estimating the
functions they need for managerial decision
making

75. Reference:
Managerial Economics fifth edition
By Charles Maurice and Christopher R. Thomas