Exam%20Feb%2028:%20sets%201,2

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Exam Feb 28 sets 1,2

• Set 1 due Thurs
• Memo C-1 due Feb 14
• Free tutoring will be available next week
  Plan A MW 4-6PM OR
  Plan B TT 2-4PM
  VOTE for
  Plan A or Plan B
  Announce results Thurs

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Kinderman Supplement

• Ch 2 Multiple Regression
• Ch 3 Analysis of Variance

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MULTIPLE REGRESSION

• Kinderman, Ch 2

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Example

• Reference Statistics for Managers
• By Levine, David M Berenson Stephan
• Second edition (1999)
• Prentice Hall

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Y dependent variable heating oil sales (gal)

• X1 Temperature (degrees)
• X2 Insulation (inches)
• X1 and X2 are independent variables
• Y bo b1X1 b2X2
• Enter data to Excel
• NOTE If you cant find Data Analysis, try
  Add-Ins

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Y 562 5X1 20X2

• Bottom table
• Coefficient Column

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Interpret coefficients

• Intercept bo 562 If temp 0 and insulation
  0, heating oil sales 562
• b1 -5 For all homes with same insulation, each
  1 degree increase in temperature should decrease
  heating oil sales by 5 gallons
• b2 -20 For all months with same temp, each
  additional 1 inch of insulation should decrease
  sales by 20 gallons

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Categorical Variables

• X 0 or 1
• Example 0 if male, 1 if female
• Example 1 if graduate, 0 if drop out
• Example 1 if citizen, 0 if alien
• NOTE not in this fuel oil example

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Estimate sales if temp 30, insulation 6

• Y 562 -5(30) 20(6) 292 gal

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Standard Error 26Top table

• Interpret Typical fuel oil sales were about 26
  gal away from average fuel oil sales of other
  homes with same temp and insulation

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COEFFICIENT OF MULTIPLEDETERMINATION

• Top table, R square
• Interpret 96 of total variation in fuel oil
  sales can be explained by variation in
  temperature and insulation

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Is there a relationship between all independent
variables and dependent variables?

• Ho Null hypothesis All coefficients 0
• Ho NO Relationship
• H1 Alternative hypothesis At least one
  coefficient is not zero
  H1 There is a relationship

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Computer output Sample data

• Hypotheses Population parameters
• Ho Parameters 0, but sample data makes it
  appear that there is a relationship
• Simple regression Ho zero slope vs H1
  slope positive or slope negative

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Exponents

• 10-1 0.1
• 10-2 0.01

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Decision Rule

• Reject Ho if Significance F lt alpha
• Middle table
• Fuel oil example Significance F 1.6E-09
• Excel E Exponent
• 1.6E-09 1.610-9 0.0000000016
• Approaches zero as limit

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Significance Fp-value

• Excel uses p-value only if t distribution
• Significance F probability F is greater than
  Sample F

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Assume alpha .05

• Since 0 lt .05, reject Ho
• We conclude there IS a relationship between fuel
  oil sales and the independent variables

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Which independent variables seem to be important
factors?

• Ho Temperature not important factor
• H1 Temperature is important
• Reject Ho if p-value lt alpha
• Bottom table p-value column, X1 row
• P-value 1.6E-09, or zero
• Reject Ho
• Temp is important

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Insulation

• Ho insulation unimportant
• H1 insulation important
• P-value 1.9E-06, or zero
• Reject Ho
• Insulation important

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Analysis of Variance (ANOVA)

• Kinderman, Ch 3

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X number of auto accidents
Live in City Live in Suburb Live in rural
1 2 1
3 0 0
2 1 0


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Hypothesis Testing

• Ho µ1 µ2 µ 3
• H1 Not all means are
• H1 There are differences among 3 populations
• H1 Average number of accidents different
  depending on where you live

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This course manual calculations

• If you used computer software, you could have as
  many populations as needed
• Homework, exam 3 populations
• Computer 4 or more populations
• Ex Ethnic classifications at CSUN

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Sample Sizes

• Column 1 n1 number of drivers sampled from
  policyholders living in city 3
• Column 2 n2 sampled from suburban drivers 3
• Col 3 n3 sampled from rural 3
• Number of rows of data
• Kinderman example Different sample sizes

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n n1 n2 n3

• n 3 3 3 9

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X number of auto accidents
Live in City Live in Suburb Live in rural
1X11 2 1
3X21 0 0
2X31 1 0


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Do not assume n13 on exam
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X number of auto accidents
Live in City Live in Suburb Live in rural
1X11 2 1
3X21 0 0
2X31 1 0
S6 S3 S1
Sample mean2 Sample mean1 Sample mean.3
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Hypotheses

• Ho Differences in sample means due to chance,
  but no differences if ALL drivers were included
  (Prop 103)
• H1 Population means are different because city
  drivers have more accidents

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Grand mean 1.1
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SSB Sum of Squares Between

• Between 3 groups
• Explained Variation
• Here Variation in number of accidents explained
  by where you live (city, suburb, rural)
• If where you live did not affect accidents, we
  would expect SSB 0
• Next slide SSB formula

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X number of auto accidents
Live in City Live in Suburb Live in rural
1X11 2 1
3X21 0 0
2X31 1 0
S6 S3 S1
Sample mean2 Sample mean1 Sample mean.3
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This example

• SSB 3(2-1.1)23(1-1.1)2 3(.3-1.1)2 4.2

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MSB Mean Square Between

• MSB SSB/2
• Note OK for this course, but bigger problems
  would have bigger denominator
• MSB 4.2/2 2.1

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SSE Sum of Squared Error

• Variation within group
• Ex Variation within group of city drivers
• Unexplained variation
• If every city driver had same number of
  accidents, we would expect SSE 0
• Formula on next slide

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X number of auto accidents
Live in City Live in Suburb Live in rural
1X11 2 1
3X21 0 0
2X31 1 0
S6 S3 S1
Sample mean2 Sample mean1 Sample mean.3
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(1-2)2 (3-2)2 (2-2)2 (2-1)2 (0-1)2
(1-1)2 (1-.3)2 (0-.3)2 (0-.3)2

• 4.67

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MSE Mean Square Error

• Mean Square Within
• Next slide is formula for this course.
• Bigger problems have bigger denominator

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MSE 0.78
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F RATIO

• Sample F statistic
• Test statistic
• SAM F

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Sam F 2.7

• Extreme case1 Where you live does not affect
  number of accidents, so SSB 0, so MSB 0, so
  sam F 0
• Extreme case 2 Every city driver has same
  number of accidents, etc, so SSE 0, so MSE 0,
  so sam F is very large

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Critical F cr F

• F table at end of Kinderman Supplement
• Appendix A, Table A.3, p 60 in Second Edition
  (assumes alpha .05)
• Column 2 (denominator of MSB)
• Row n 3 (denominator of MSE)
• Correct for this course, different for bigger
  problems

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Example

• Col 2
• Row 9-3 6
• Cr F 5.14

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Hypothesis Testing

• Ho µ1 µ2 µ 3
• H1 Not all means are
• H1 There are differences among 3 populations
• H1 Average number of accidents different
  depending on where you live

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Decision Rule

• Reject Ho if sam F gt cr F
• Only right tail since SSBgt0, SSEgt0, so sam Fgt0
• If you reject Ho, you conclude that where you
  live affects number of accidents
• If you do not reject Ho, you conclude that there
  is too much variation within city drivers, etc to
  draw any conclusions

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Example

• Since 2.7 is NOT gt 5.14, we can NOT reject Ho
• Differences between city and suburb, etc are NOT
  significant

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Computer Approach

• Similar to multiple regression
• Reject Ho if Significance F lt alpha
• Needed if more than 3 groups