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summarize the key economic principles and concepts in each of the assigned chapters.

Chapter 3
Marginal Analysis for
Optimal Decisions
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
1-1
Learning Objectives
❖ Define several key concepts and terminology related to
marginal analysis
❖ Use marginal analysis to find optimal activity levels in
unconstrained maximization problems and explain why
sunk costs, fixed costs, and average costs are irrelevant
for decision making
❖ Employ marginal analysis to find the optimal levels of
two or more activities in constrained maximization and
minimization problems
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-2
Optimization
❖ A persons decision is optimal if it leads to
the best outcome under a given set of
circumstances. This is accomplished by
using Marginal Analysis.
❖ A person needs to determine the benefit
of a changing activity and compare it with
the cost associated with the change in
activity.
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-3
Optimization
❖ Tactical Decisions
❖ Strategic Decisions
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3-4
Optimization
❖ An optimization problem involves the
specification of three things:
~ Objective function – what is to be maximized
or minimized (profit, satisfaction, value).
~ Activities or choice variables that determine
the value of the objective function –
production level for profits.
~ Any constraints that may restrict the values of
the choice variables – such as cost.
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3-5
Optimization
❖ We have discrete and continuous choice
variables.
❖ A Discrete Choice Variable can only take
on specific integer values.
❖ A Continuous Choice Variable can take
on any value between two points.
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-6
Optimization
❖ Maximization problem
~ An optimization problem that involves
maximizing the objective function
❖ Minimization problem
~ An optimization problem that involves
minimizing the objective function
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3-7
Optimization
❖ Unconstrained optimization
~ An optimization problem in which the decision
maker can choose the level of activity from an
unrestricted set of values.
❖ Constrained optimization
~ An optimization problem in which the decision
maker chooses values for the choice
variables from a restricted set of values (such
as total costs).
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-8
Choice Variables
❖ Activities or choice variables determine
the value of the objective function
❖ Discrete choice variables
~ Can only take specific integer values
❖ Continuous choice variables
~ Can take any value between two end points
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-9
Marginal Analysis
❖ Analytical technique for solving
optimization problems that involves
changing values of choice variables by
small amounts to see if the objective
function can be further improved
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-10
Net Benefit
❖ Net Benefit (NB)
~ Difference between total benefit (TB) and total
cost (TC) for the activity
~ NB = TB – TC
❖ Optimal level of the activity (A*) is the
level that maximizes net benefit
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-11
Optimization
❖ The optimal level of activity does not
generally result in the maximization of
benefits.
❖ The optimal level of activity in an
unconstrained maximization problem
does not result in the minimization of total
costs.
❖ Only marginal benefits and marginal costs
are relevant in the decision process.
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-12
Optimal Level of Activity
(Figure 3.1)
Total benefit and total cost (dollars)
TC
4,000
D
B
•
2,310
F
•
• D’
3,000
•
•
G
TB
2,000
NB* = $1,225
C
•
1,085
1,000
• B’
•C’
0
200
A
350 = A*
600
700
1,000
Level of activity
Net benefit (dollars)
Panel A – Total benefit and total cost curves
M
1,225
1,000
•c’’
•
•
600
0
d’’
200
350 = A*
•
600
Level of activity
A
f’’
1,000
NB
Panel B – Net benefit curve
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-13
Marginal Benefit & Marginal Cost
❖ Marginal benefit (MB)
~ Change in total benefit (TB) caused by an
incremental change in the level of the activity
❖ Marginal cost (MC)
~ Change in total cost (TC) caused by an
incremental change in the level of the activity
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-14
Marginal Benefit & Marginal Cost
Change in total benefit TB
MB =
=
Change in activity
A
Change in total benefit TC
MC =
=
Change in activity
A
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-15
Using Marginal Analysis to Find
Optimal Activity Levels
❖ If marginal benefit > marginal cost
~ Activity should be increased to reach highest
net benefit
❖ If marginal cost > marginal benefit
~ Activity should be decreased to reach highest
net benefit
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-16
Using Marginal Analysis to Find
Optimal Activity Levels
❖ Optimal level of activity
~ When no further increases in net benefit are
possible
~ Occurs when MB = MC
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
3-17
Unconstrained Maximization with
Discrete Choice Variables
❖ Increase activity if MB > MC
❖ Decrease activity if MB < MC ❖ Optimal level of activity ~ Last level for which MB exceeds MC ~ To make the optimal decision for a discrete choice variable, decision makers must increase activity until the last level of activity is reached for which marginal benefit exceeds marginal cost. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 3-18 Irrelevance of Sunk, Fixed, and Average Costs ❖ Sunk costs ~ Previously paid & cannot be recovered ❖ Fixed costs ~ Constant & must be paid no matter the level of activity ❖ Average (or unit) costs ~ Computed by dividing total cost by the number of units of activity © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 3-19 Irrelevance of Sunk, Fixed, and Average Costs ❖ Decision makers wishing to maximize the net benefit of an activity should ignore these costs, because none of these costs affect the marginal cost of the activity and so are irrelevant for optimal decisions © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 3-20 Constrained Optimization ❖ The ratio MB/P represents the additional benefit per additional dollar spent on the activity ❖ Ratios of marginal benefits to prices of various activities are used to allocate a fixed number of dollars among activities ❖ It is marginal benefit per dollar spent that matters in decision making. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 3-21 Constrained Optimization ❖ To maximize or minimize an objective function subject to a constraint ~ Ratios of the marginal benefit to price must be equal for all activities ~ Constraint must be met MBA MBB MBC MBZ = = ... = PA PB PC PZ © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 3-22 Summary ❖ Marginal analysis is an analytical technique for solving optimization problems by changing the value of a choice variable by a small amount to see if the objective function can be further improved ❖ The optimal level of the activity (A*) is that which maximizes net benefit, and occurs where marginal benefit equals marginal cost (MB = MC) ~ Sunk costs have previously been paid and cannot be recovered; Fixed costs are constant and must be paid no matter the level of activity; Average (or unit) cost is the cost per unit of activity; these 3 types of costs are irrelevant for optimal decision making ❖ The ratio MB/P denotes the additional benefit of that activity per additional dollar spent (“bang per buck”) © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 3-23 Chapter 4 Basic Estimation Techniques © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 1-1 Learning Objectives ❖ Set up and interpret simple linear regression equations ❖ Estimate intercept and slope parameters of a regression line using the method of least‐squares ❖ Determine statistical significance using either t‐tests or p‐values associated with parameter estimates ❖ Evaluate the “fit” of a regression equation to the data using the R2 statistic and test for statistical significance of the whole regression equation using an F‐test ❖ Set up and interpret multiple regression models ❖ Use linear regression techniques to estimate the parameters of two common nonlinear models: quadratic and log‐linear regression models © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-2 Basic Estimation ❖ Parameters ~ The coefficients in an equation that determine the exact mathematical relation among the variables – for the cost function to be useful for decision making we must know the values of the parameters. ❖ Parameter estimation ~ The process of finding estimates of the numerical values of the parameters of an equation © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-3 Regression Analysis ❖ Regression analysis ~ A statistical technique for estimating the parameters of an equation and testing for statistical significance. ~ Regression analysis uses data on economic variables to determine a mathematical equation that describes the relationships between the economic variables. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-4 Regression Analysis ❖ Dependent variable ~ Variable whose variation is to be explained ❖ Explanatory variables ~ Variables that are thought to cause the dependent variable to take on different values © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-5 Simple Linear Regression ❖ True (Actual) regression line relates dependent variable Y to one explanatory (or independent) variable X Y = a + bX ~ Intercept parameter (a) gives value of Y where regression line crosses Y-axis (value of Y when X is zero) ~ Slope parameter (b) gives the change in Y associated with a one-unit change in X: b = Y X © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-6 Regression Analysis ❖ If Y = 2500 + 10X ❖ Then ❖ Y will change on average 10 units for every 1 change in X © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-7 Simple Linear Regression ❖ Regression line shows the average or expected value of Y for each level of X ❖ True (or actual) underlying relation between Y and X is unknown to the researcher but is to be discovered by analyzing the sample data ❖ Random error term ~ Unobservable term added to a regression model to capture the effects of all the minor, unpredictable factors that affect Y but cannot reasonably be included as explanatory variables © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-8 Fitting a Regression Line ❖ The purpose of a regression analysis is: ❖ 1. To estimate the parameters (a and b) of the true (actual) regression line (population regression line). ❖ 2. to test the estimated values of the parameters to see if they are statistically significant. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-9 Fitting a Regression Line ❖ Estimating “a” and “b” is equivalent to fitting a straight line through a scatter of data points on a graph. ❖ The data collected is on both the dependent and explanatory variables. ❖ The objective of regression analysis is to find the straight line that best fits the scatter of data points (sample regression line). © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-10 Fitting a Regression Line ❖ Time series ~ A data set in which the data for the dependent and explanatory variables are collected over time for a single firm ❖ Cross-sectional ~ A data set in which the data for the dependent and explanatory variables are collected from many different firms or industries at a given point in time © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-11 Fitting a Regression Line ❖ The sample regression line is only an estimate of the true regression line. Naturally, the larger the size of the sample, the more accurately the sample regression line will estimate the true regression line. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-12 Fitting a Regression Line ❖ Method of least squares – used for Regression Analysis ~ A method of estimating the parameters of a linear regression equation by finding the line that minimizes the sum of the squared distances from each sample data point to the sample regression line (you want as tight a fit as you can get). © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-13 Fitting a Regression Line ❖ Parameter estimates are obtained by choosing values of a & b that minimize the sum of squared residuals ~ The residual is the difference between the actual and fitted/predicted values of Y: Yi – Ŷi ~ Equivalent to fitting a straight line through a scatter diagram of the sample data points – in order to minimize the sum of the squared residuals. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-14 Fitting a Regression Line ❖ The sample regression line is an estimate of the true (or population) regression line ˆ Yˆ = aˆ + bX ~Where â and b̂ are least squares estimates of the true (population) parameters a and b © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-15 Sample Regression Line (Figure 4.2) S Sales (dollars) 70,000 S 60,000 Sii = = 60,000 60,000 ei 50,000 20,000 10,000 • • 40,000 30,000 • Sample regression line Ŝi = 11,573 + 4.9719A • = 46,376 Ŝi Ŝ =i 46,376 • • • A 0 2,000 4,000 6,000 8,000 10,000 Advertising expenditures (dollars) © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-16 Unbiased Estimators ❖ The estimates â & b̂ do not generally equal the true values of a & b ~ â & b̂ are random variables computed using data from a random sample ❖ The distribution of values the estimates might take is centered around the true value of the parameter ❖ An estimator is unbiased if its average value (or expected value) is equal to the true value of the parameter © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-17 Unbiased Estimators ❖ We now turn to the task of testing hypotheses about true (actual) values of “a” and “b”, which are unknown to the researcher, using information contained in the sample to see if there is a significant relationship or simply a relationship based on randomness. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-18 Statistical Significance ❖ Do the coefficients mean anything? Do they statistically explain the change in the dependent variable? ❖ The farther away from 0 (+ or -) the estimated coefficient is, the estimated coefficient is considered Statistically Significant. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-19 Statistically Significant ❖ If Y is indeed related to X, the true value of the slope of the parameter will be either a positive or negative number. ❖ Remember, if the coefficient is the change in Y divided by the change in X, and the coefficient is zero, then no change in Y will occur when there is a change in X. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-20 Statistically Significant ❖ We need to test for the statistically significance of a parameter because we do not know the true value of the parameters. i.e. they are a random sample. ❖ Because of randomness in sampling, different samples will result in different values. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-21 Statistically Significant ❖ The estimator of the coefficient is Unbiased if the average value of the estimator is equal to the true value of the parameter. ❖ Unbiased does not mean that any one estimate of the coefficient is close to the true value. Unbiased means that in repeated samples, the estimates tend to be centered around the true value. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-22 Statistical Significance ❖ Statistical significance ~ There is sufficient evidence from the sample to indicate that the true value of the coefficient is not zero ❖ Hypothesis testing ~ A statistical technique for making a probabilistic statement about the true value of a parameter © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-23 Statistical Significance ❖ Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e., b  0) ❖ Even if b = 0, it is possible that the sample will produce an estimate b̂ that is different from zero (Type I error) ❖ Test for statistical significance using t-tests or p-values © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-24 Statistical Significance ❖ A t-test is used to make a probabilistic statement about the likelihood that a true parameter value of a coefficient is not equal to zero. ❖ The t-ratio indicates how much confidence one can have that the true value of a coefficient is actually larger than zero. The larger the value the better. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-25 Statistical Significance ❖ First determine the level of significance ~ Probability of finding a parameter estimate to be statistically different from zero when, in fact, it is zero ~ Probability of a Type I Error ❖ 1 – level of significance = level of confidence ~ Level of confidence is the probability of correctly failing to reject the true hypothesis that b = 0 © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-26 Statistical Significance ❖ The level of significance of a t-test is the probability that the test will indicate the coefficient is not equal to zero when in fact it is equal to zero. ❖ For example a level of significance of .01, .02, .05 assumes a willingness to accept a 1, 2, or 5 percent probability of finding a parameter to be significant when it is not (99%, 98% or 95% confidence level). © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-27 Performing a t-Test b̂ ❖ t-ratio is computed as t = Sb̂ where Sb̂ is the standard error of the estimate bˆ ❖ Use t-table to choose critical t-value with n – k degrees of freedom for the chosen level of significance ~ n = number of observations ~ k = number of parameters estimated © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-28 Performing a t-Test ❖ t-statistic ~ Numerical value of the t-ratio ❖ If the absolute value of t-statistic is greater than the critical t, the parameter estimate is statistically significant at the given level of significance © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-29 Using p-Values ❖ Treat as statistically significant only those parameter estimates with p-values smaller than the maximum acceptable significance level ❖ p-value gives exact level of significance ~ Also the probability of finding significance when none exists ~ One minus the p-value is the exact degree of confidence that can be assigned to a particular parameter estimate. © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-30 Coefficient of Determination ❖ R2 measures the fraction of total variation in the dependent variable (Y) that is explained by the variation in X ~ Ranges from 0 (Explains none) to 1 (explains all) ~ High R2 indicates Y and X are highly correlated, and does not prove that Y and X are causally related © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-31 F-Test ❖ Used to test for significance of overall regression equation (explained and unexplained variations) ❖ Compare F-statistic to critical F-value from F-table ~ Two degrees of freedom, n – k & k – 1 (# of independent variables) ❖ If F-statistic exceeds the critical F, the regression equation overall is statistically significant at the specified level of significance © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-32 Multiple Regression ❖ Uses more than one explanatory variable ❖ Coefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable, all else constant © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-33 Summary ❖ A simple linear regression model relates a dependent variable Y to a single explanatory variable X ~ The regression equation is correctly interpreted as providing the average value (expected value) of Y for a given value of X ❖ Parameter estimates are obtained by choosing values of a and b that create the best-fitting line that passes through the scatter diagram of the sample data points ❖ If the absolute value of the t-ratio is greater (less) than the critical t-value, then is (is not) statistically significant ~ Exact level of significance associated with a t-statistic is its p-value ❖ A high R2 indicates Y and X are highly correlated and the data tightly fit the sample regression line © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-34 Summary ❖ If the F-statistic exceeds the critical F-value, the regression equation is statistically significant ❖ In multiple regression, the coefficients measure the change in Y associated with a one-unit change in that explanatory variable ❖ Quadratic regression models are appropriate when the curve fitting the scatter plot is U-shaped or ∩-shaped (Y = a + bX + cX2) ❖ Log-linear regression models are appropriate when the relation is in multiplicative exponential form (Y = aXbZc) ~ The equation is transformed by taking natural logarithms © 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 4-35 Purchase answer to see full attachment

  
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