Class Notes Sessions5-6

8/14/2019 Class Notes Sessions5-6

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One Way ANOVA

ANOVA stands for Analysis of Variance

ANOVA allows us to compare the means from more than twosets of scores.

A significant ANOVA indicates that changes in the independentvariable affect the dependent variable.

ANOVA does not indicate which pairs of conditions aresignificantly different.

Use planned contrasts or unplanned (post hoc) contrasts toassess whether pairs of conditions are significantly different.

ANOVA Assumptions

1. Normally distributed populations2. Equal population variances3. Random sampling used4. Dependent variable uses an interval or ratio scale


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Digression on Scales: Levels of Measurement

Interval: 0 doesnt mean none (e.g., IQ score)

- distances between points on scale are equal but ratiosarent meaningful (e.g., temperature)

Ratio: Same as interval scale, but 0 means none andratios are meaningful (e.g., weight or age: a personwho is 50 is twice as old as one who is 25).

Nominal: Numbers are just labels for attributes (e.g., color)

Ordinal: categories have a logical order (e.g., ranks)

Digression on Scales: Types of Data

Continuous: Numerical data that can be fractional (e.g.,

temperature)

Discrete: Numerical data that cannot be fractional (e.g.,number of World Cup trophies)

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One-way ANOVANational Airlines

National Airlines recently introduce a daily early-morning nonstop flightbetween Houston and Chicago. The vice president of marketing forNational Airlines decided to perform a statistical test to see whetherNationals average passenger load on this new flight is different fromthat of each of its two major competitors (which we will call competitor1 and competitor 2). Ten early-morning flights were selected at randomfrom each of the three airlines and the percentage ofunfilledseats oneach flight was recorded. These data are stored in an EXCEL file on thewebsite at National Airlines (Excel).

Is there evidence that Nationals average passenger load on the newflight is different from that of its two competitors? Report a p value andinterpret the results of the statistical test.

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Raw Data (in Excel)

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Raw Data (in SPSS)

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Transform Data into Analysis-Ready Form

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Analyze Compare Means One-Way ANOVA

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Post Hoc Contrasts

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Results

Descriptives

Unfilled

10 9.80 2.044 .646 8.34 11.26 7 13

10 11.30 2.003 .633 9.87 12.73 7 13

10 12.60 2.011 .636 11.16 14.04 9 15

30 11.23 2.269 .414 10.39 12.08 7 15

National

Competitor 1

Competitor 2

Total

N Mean Std. Deviation Std. Error Lower Bound Upper Bound

95% Confidence Interval for

MeanMinimum Maximum

ANOVA

Unfilled

39.267 2 19.633 4.815 .016

110.100 27 4.078

149.367 29

Between Groups

Within Groups

Total

Sum of

Squares df Mean Square F Sig.

Post Hoc Tests

Multiple Comparisons

Dependent Variable: Unfilled

Bonferroni

-1.500 .903 .325 -3.81 .81

-2.800* .903 .013 -5.11 -.49

1.500 .903 .325 -.81 3.81

-1.300 .903 .484 -3.61 1.01

2.800* .903 .013 .49 5.11

1.300 .903 .484 -1.01 3.61

(J) Airline

Competitor 1

Competitor 2

National

Competitor 2

National

Competitor 1

(I) Airline

National

Competitor 1

Competitor 2

Mean

Difference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

The mean difference is significant at the .05 level.*.

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The 2 Test

For data in which each outcome is assigned to a single (and only

one) mutually exclusive and exhaustive category

Derived from square of Z statistic

Assumes: Independence of observations (can't take severalobservations from 1 person and analyze with 2 )

Compares the observed and expected values

Needs at least 5 expectedobservations per cell

Values range from 0 on up (no negative values)

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One-way 2

Are There Differences Among theLevels of 1 Variable?

H0: P1 = P2 = P3 = P4 =4

1

(where P1 + P2 + P3 + P4 = 1)

Ha: At least one Population Proportion 41

E

)E-(O=

22

E)E-O(+. . .+

E)E-O(=

m

mm

2

1

11

2

2

Where Oi and Ei are the observed and expected # ofoccurrences for m (exhaustive & mutually exclusive)outcomes

Squared deviations in which large disparities count formore than small disparities

df = (# levels in the independent variable 1)

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100 Analysts Rate an IPO

Strong Buy Buy Hold Sell Strong Sell

24 33 22 16 5H0 : PSB = PB = PH = PS = PSSHa : PSB PB PH PS PSS

Actual & Expected ( )

Strong Buy Buy Hold Sell Strong Sell

24(20) 33(20) 22(20) 16(20) 5(20)

Critical Value (df = 4, )05.= : 49.9)4(2 =

=E

EO2

2 )( = 50.21

20

15

20

4

20

2

20

13

20

422222

=++++

Reject H0 because the test statistic (21.50) is greaterthan the critical value (9.49).

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100 Analysts Rate an IPO:Testing Unequal Categories

Test the null hypothesis that twice as many willoffer some form of buy recommendation (eitherStrong Buy or Buy) than will offer either a hold orsome form of sell recommendation (Sell or StrongSell).

Collapse analysts recommendations into 3categories:

Buy Hold Sell

57 22 21

H0 : PB = 2PH = 2PSHa : PB 2PH 2Ps

Critical Value ( )05.= : 99.5)2(2 =

Actual & Expected ( )Buy Hold Sell

57(50)

22(25)

21(25)

=EEO

2

2 )( 98.125

16

25

9

50

492=++=

Do not reject H0 because the test statistic (1.98) isless than the critical value (5.99).

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100 Analysts Rate an IPO:Testing Unequal Categories in SPSS

Analyze Nonparametric Chi-Square

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Set up the expected values for each category

Results

Analysts' Recommendations

57 50.0 7.0

22 25.0 -3.0

21 25.0 -4.0

100

Buy

Hold

Sell

Total

Observed N Expected N Residual

Test Statistics

1.980

2

.372

Chi-Squarea

df

Asymp. Sig.

Analysts' Recommendations

0 cells (.0%) have expected frequencies less than

5. The minimum expected cell frequency is 25.0.

a.

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Two-way 2

Are There Differences Between 2 Variables?

H0: Variables A and B are independentHa: Variables A and B are dependent

df = (# rows - 1)(# columns - 1)

Tests nondirectional hypotheses only, using a single tail

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SPSS: 2-Way 2Tests

Vioxx data file:

Industry [Industry ties? 1=no, 2=yes]Vioxx [Bring Vioxx back? 1=no, 2=yes]

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Analyze Descriptives Crosstabs

Click on Statistics; select chi-square

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Click OK. This is the 2output:Crosstabs

Case Processing Summary

32 100.0% 0 .0% 32 100.0%Industry Ties? *

Bring Vioxx Back?

N Percent N Percent N Percent

Valid Missing Total

Cases

Industry Ties? * Bring Vioxx Back? Crosstabulation

Count

14 8 22

1 9 10

15 17 32

no

yes

Industry

Ties?

Total

no yesBring Vioxx Back?

Total

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Notice that the 2 test below is significant (p = .005), but notentirely reliable because the expected cell count in one of thecells is less than 5. (You should be able to verify that the lower

left cell in the Crosstabulation above is the one with theundercount.)

Chi-Square Tests

7.942b 1 .005

5.935 1 .015

8.893 1 .003

.007 .006

7.694 1 .006

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Pearson Chi-Square

Continuity Correctiona

Likelihood Ratio

Fisher's Exact Test

Linear-by-Linear

Association

N of Valid Cases

Value df

Asymp. Sig.

(2-sided)

Exact Sig.

(2-sided)

Exact Sig.

(1-sided)

Computed only for a 2x2 tablea.

1 cells (25.0%) have expected count less than 5. The minimum expected count is 4.

69.

b.

When one or more of your cells has an expected count less than5, report Fisher's Exact Test (in the SPSS output). FishersExact Test has no test statistic, no critical value, and no

confidence interval. Report it as follows: p = .007, FishersExact Test, 2-tailed.

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Correlation

How do the scores on one variable change with the scores on

another variable?

Correlations are concerned with measuring the direction andmagnitude of a linear relationship between two variables.

The stronger the correlation, the more accurately we can predictY from knowing X.

Scatterplot: A graph containing clusters of dots that represent allX-Y pairs of observations.

Involves an examination of pairs of X-Y scores (one-sampleprocedure).

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Correlation Coefficients

Measures extent to which individual Xi-Yi scores that make up a

pair occupy the same or opposite positions within theirdistributions.

- Pos relation: Pairs tend to occupysimilarrelativepositions in their distributions

- Neg relation: Pairs tend to occupy opposite relativepositions in their distributions

Two types (there are others as well):- Pearson r (continuous data): rxy

- Phi Coefficient (binary variables: 2 X 2 Tables):

Range from -1 to 11 = perfect pos relation

-1 = perfect neg relation0 = No relation

Failure to find strong r may mean:(a) chance(b) variables are unrelated, or(c) the variables are related nonlinearly.

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R Computation (by hand)

1. Transform each Y score into a Z score (Zy)

2. Transform each X score into a Z score (Zx)3. Determine correspondence between each of the paired Zs

- r indicates the average correspondence between the pairedZs.

r = Mean of the crossproduct of Z scores.

Population Sample

N

ZZ=r

yx

1

N

ZZ=r

yx

(note: Zs will differ for population & samples because thedenominator for computing population Zs is and the

denominator for computing sample Zs is s.)

When large pos correspondence: Z crossproduct is pos. & large

When small neg correspondence: Z crossproduct is neg & small- (lots of + and - canceling each other out)

Strength of Relationship

r2 = Proportion of variability of Y accounted for by X

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Strong Correlation

(population computation)

Student # High School # College Zx Zy

As (X) As (Y)

Alejandro 13 14 1.500.50Bernardo 9 18 0.50 1.50Carlos 7 12 0.00 0.00

Dominique 5 10 -0.50 -0.50Enrique 1 6 -1.50 -1.50

80.05

)5.1)(5.1()5.)(5.()0)(0()5.1)(5(.)5)(.5.1( =

++++=

=

N

ZZr

yx

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Strong Correlation: Using SPSS

Analyze Correlate Bivariate

Correlation Output

Correlations

1 .800

. .104

5 5.800 1

.104 .

5 5

Pearson Correlation

Sig. (2-tailed)

NPearson Correlation

Sig. (2-tailed)

N

High School

College

High School College

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Two Points of Caution with Correlations1. Restriction of range (i.e., truncated range) problem

When the relevant range of X or Y scores is a truncated part ofwhole, then the truncated X-Y correlation will be smaller thanthe whole X-Y correlation.

2. Correlation does not mean causation

- may be a correlated 3rd variable

- Even if no 3rd variable is involved, its not always clearwhich variable is the cause and which is the effect.

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Phi Coefficient

Correlation for Categorical Data (2 X 2 Tables):

a b

c d

d)+c)(b+d)(a+b)(c+(a

bc-ad=

Yes No50 20

10 40=

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Phi Coefficient (using SPSS)

10 5

5 8

Analyze Descriptive Statistics Crosstabs Statistics

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Click Statistics and check Phi and Cramers V

Symmetric Measures

.282 .136

.282 .136

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Phi

Cramer's V

Nominal by

Nominal

N of Valid Cases

Value Approx. Sig.

Not assuming the null hypothesis.a.

Using the asymptotic standard error assuming the null

hypothesis.

b.

(ignore Cramers V)

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Regression

Regression: The primary purpose of regression is prediction

Predictions about the linear relationship between independent anddependent variables.

Independent = predictor = explanatoryDependent = response = criterion

Types of Regression

1. Linear (least squares regression line)Simple regression: one predictor variableMultiple regression: multiple predictor variables

2. Nonlinear (can linearize many of these via transformation)- Positive curvilinear (e.g., diminishing marginal utility)- Polynomial (quadratic parabola-shaped; cubic)- Exponential or negative curvilinear (L-shaped)

3. Logistic (when dependent variable is categorical)- Example: graduate or not; sales are weak/moderate/strong

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Lines: Y = bo + b1X

bo and b1 are regression coefficients- can be positive or negative

- b1 is more important than b0

bo = Y intercept (value of Y when X=0)b1 = Slope (how much Y changes when X changes by 1 unit)

Example #1: Suppose Aeromexico wants to examine the relationbetween number of flight delays and number of passenger complaints.

X = Number of flight delaysY = Number of passenger complaints.

Suppose that the data are as follows (X, Y):(0, 1), (1, 3), (2, 5)

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Scatterplot: Delays vs. Complaints

2.001.501.000.500.00

delays

5.00

4.00

3.00

2.00

1.00

complaints

The line that fits these dataperfectly is: Y = 1 + 2X# Complaints = 1 + 2 (# Flight Delays)

X Y0 1 + 2(0) = 1 (0,1)1 1 + 2(1) = 3 (1,3)

2 1 + 2(2) = 5 (2,5)

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The regression line (also called Least squares regression line)minimizes the squared difference between the observed and predictedvalues of the response variable (as give by the regression line).

- The difference between the actual and predicted values is calledthe residual.

- Minimizing these squared residuals gives slope that is as close aspossible to true slope.

10.008.006.004.002.000.00

delays

20.00

15.00

10.00

5.00

0.00

complaints

R Sq Linear = 0.475

(Well talk about what R Sq Linear means later on)

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Example #2: Suppose UT wants to examine relation between alumnidonations to the school and number of football victories.

X = Number of football victoriesY = Amount of alumni donations the following year

Alumni Donations = $10,000,000 + $200,000 (# Football Victories)

Y = 10,000,000 + 200,000X

Caution #1: X-Y relation may not be causal

Caution #2: Regression line estimates most trustworthy near bulk ofdata (usually center).

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Linear Regression Assumptions

1. Linearity- Linear relationship between X and Y

- Same expected change in Y moving from X1 to X2 vs.moving from X2 to X3

Test: X-Y Scatterplot (look for nonlinearities)

Correction: Insert curvilinear term (Usually quadratic: X2)Y = bo + b1X + b2X

2

: Log transformation of X variable (if data are positive)- brings large values down, pushes small values furtherapart

2. Independence of Observations- Residuals across Xs are not correlated

Test: Durbin-Watson (0-4) (tests residual correlation among Xs)0.0 - 1.5 = pos correlation1.5 - 2.5 = no correlation2.5 - 4.0 = negative correlation

Correction: Transformation of Y variable (percentages or logs)

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3. Normality- The distrib. at each Xi is normal- The errors have normal distribution

Test #1: Plot histogram of residuals (should be normal)

Test #2: Normal Prob. Plot- Plot of cumulative probabilities- Should follow diagonal (if residuals follow normal distrib)

Correction: Log transformation of Y

4. Constant Variance (homoskedasticity)- Each Yi distrib. has same variance- Means that effects of other factors does not depend on level of X.- Common problem: Variance up as X increases (funnel shape)

Test: Scatterplot of X vs. Residuals- Should not show funnel shape pattern

Correction: Log transformation of Y

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Example: Simple Linear Regression

Houston Astros Payroll

Identify a regression equation that predicts the median salary for aHouston Astros baseball player based on knowledge of the total team

payroll

Independent variable: Total PayrollDependent variable: Median Salary

Here are your data (figures are in thousands)

You can access this data file on the website as well (Houston Astrossalary data)

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1. Create XY Scatterplot

Graphs Scatter Simple Define OK

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Median Salary Total Payroll Scatterplot

10000.00 20000.00 30000.00 40000.00 50000.00 60000.00 70000.00 80000.00

Total Payroll

0.00

300.00

600.00

900.00

1200.00

1500.00

MedianSalary

This scatterplot shows that the linearity assumption is OK- well check the other 3 assumptions shortly

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2. Visual check for outliers (remove if necessary)

3. Add regression line:Double click on graph

Single click on a data point (it will enlarge and change color)

Elements Fit Line At Total

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Fit Line at Total Linear

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4. Conduct Regression Analysis

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Put Independent and Dependent variables in the right boxes

Click Statistics

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Click Plots

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Click Save

By checking these boxes, you will create extra columns on your datafile. You will get a Predicted Values (PRE_1) column and a ResidualValues (RES_1) column.

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5. Examine Regression Output

M o d e l S u m m a ryb

.754a .56 9 .5 40 2 20 .5 39 7 8 .56 9 1 9 .79 0 1 1 5 .0 00 2

M o d e l

1

R R Sq ua re

Ad jus ted

R Squa re

Std . Er ro r o f

the Es t imate

R Squa re

C han ge F C h a ng e d f1 d f2 S ig . F C h a ng e

Chan ge Sta t is t ics

Durb i

Wa t s

Pred ic to rs : (Cons tan t ) , To ta l Payro l la .

Dependent Var iab le : Med ian Sa la ryb .

ANOVAb

962530.2 1 962530.159 19.790 .000a

729566.9 15 48637.793

1692097 16

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), Total Payrolla.

Dependent Variable: Median Salaryb.

Coefficientsa

110.736 111.951 .989 .338

.012 .003 .754 4.449 .000

(Constant)

Total Payroll

Model1

B Std. Error

Unstandardized

Coefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: Median Salarya.

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6. Is model statistically significant?

Yes, because F = 19.79, p = .000 (i.e., p < .001).

7. Identify equation for the simple linear model (i.e., the regression line)

Coefficientsa

110.736 111.951 .989 .338

.012 .003 .754 4.449 .000

(Constant)

Total Payroll

Model

1

B Std. Error

Unstandardized

Coefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: Median Salarya.

Y = Y Intercept + Beta * (X)

Median Salary = 110.736 + .012 (Total Payroll)

Or, in actual dollars:Median Salary = $110,736 + .012 (Total Payroll)

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8. Check the other 3 linear regression assumptions

8a.Independence: D-W = 2.346 (OK, because its between 1.5 and 2.5)

8b.Normality: Histogram of Residual (is it normal?): Normal Prob. Plot (are points near the diagonal?)

-2 -1 0 1 2 3

Regression Standardized Residual

0

1

2

3

4

Frequency

Mean = -6.94E-17Std. Dev. = 0.968N = 17

Dependent Variable: Median Salary

Histogram

OK, because residuals have a roughly normal shape

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0.0 0.2 0.4 0.6 0.8 1.0

Observed Cum Prob

0.0

0.2

0.4

0.6

0.8

1.0

ExpectedCumProb

Dependent Variable: Median Salary

Normal P-P Plot of Regression Standardized Residual

OK, because points are near the diagonal

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8c. Constant variance?: Is there an absence of a funnel shape in scatterplot of X vs.Residuals?

Go to your Modified Data File:

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Heres a look at your data file ordered from lowest to highest payroll,where some of the columns are rearranged to make it more readable:

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Test the Constant Variance assumption by looking at theX vs. Residuals scatterplot. Check for funnel pattern.

10000.00 20000.00 30000.00 40000.00 50000.00 60000.00 70000.00 80000.00

Total Payroll

-400.00000

-200.00000

0.00000

200.00000

400.00000

600.00000

UnstandardizedResidual

Theres a hintof a bit of a funnel pattern here.(Consider a log transformation of Y variable Median)

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9. Search output for Case Diagnostics that describe outliers

None were found here, so nothing shows up in the SPSS output

But if you changed the Casewise Diagnostics (in Statistics) toshow outliers beyond 1 sd

Heres what youd get:

Casewise Diagnostics(a)

Case Number Std. Residual Median SalaryPredicted

Value Residual

3 1.060 500.00 266.1707 233.82928

8 -1.320 185.00 476.0631 -291.06310

14 2.229 1300.00 808.3513 491.64868

15 -1.558 500.00 843.7011 -343.70113

16 1.218 1200.00 931.4056 268.59437

17 -1.051 750.00 981.7387 -231.73868

a Dependent Variable: Median Salary

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Dont Trust Your Model TOO Much

Question:

The Houston Astros payroll in 2005 = $76,779.000. What does theregression line predict the median salary will be?

Answer:Predicted Median Salary =$110,736 + (.012)(76,779,000) = $1,032,084

Actual: $500,000

Question:Why was the model so far off?

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1988 Houston Astros (Total payroll = $13,455,000; Median = $500,000)T

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2005 Houston Astros (Total payroll = $76,779,000; Median = $500,000)

Class Notes Sessions5-6

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Transcript of Class Notes Sessions5-6