corelation and regression

8/12/2019 Corelation and Regression

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Elementary StatisticsLarson Farber

9 Correlation and Regression


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orrelation

Section 9.1


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Correlation

What type of relationship exists between the twovariables and is the correlation significant?

x y

Cigarettes smoked per day

Score on SATHeight

Hours of Training

Explanatory(Independent) Variable

Response(Dependent) Variable

A relationship between two variables

Number of Accidents

Shoe Size Height

Lung Capacity

Grade Point AverageIQ


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Negative Correlationasxincreases, ydecreases

x= hours of trainingy= number of accidents

Scatter Plots and Types of Correlation

60

50

40

30

20

10

0

0 2 4 6 8 10 12 14 16 18 20

Hours of Training

Accidents


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Positive Correlationasxincreases,y increases

x= SAT scorey= GPA

GPA


4.003.753.50

3.002.752.502.252.00

1.501.75

3.25

300 350 400 450 500 550 600 650 700 750 800

Math SAT


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No linear correlation

x= height y= IQ


160

150

140

130120

110

100

90

8060 64 68 72 76 80

Height

IQ


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Correlation Coefficient

A measure of the strength and direction of a linearrelationship between two variables

The range of ris from 1 to 1.

If ris close to1 there is a

strongpositive

correlation.

If ris close to 1there is a strongnegativecorrelation.

If ris close to0 there is nolinear

correlation.

1 0 1


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x y8 78

2 925 90

12 5815 43

9 746 81

AbsencesFinalGrade

Application

959085807570656055

45

40

50

0 2 4 6 8 10 12 14 16

FinalGrad

e

X

Absences


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608484648100

3364184954766561

624184450

696645666486

57 516 3751 579 39898

1 8 782 2 923 5 90

4 12 585 15 436 9 747 6 81

644

25

1442258136

xy x2

y2

Computation of r

x y


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ris the correlation coefficient for the sample. Thecorrelation coefficient for the population is (rho).

The sampling distribution for ris a t-distribution with n

2 d.f.

Standardized teststatistic

For a two tail test for significance:

For left tail and right tail to testnegative or positive significance:

Hypothesis Test for Significance

(The correlation is not significant)

(The correlation is significant)


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A t-distribution with 5 degrees of freedom

Test of Significance

You found the correlation between the number of times absentand a final grade r= 0.975. There were seven pairs of

data.Test the significance of this correlation. Use = 0.01.

1. Write the null and alternative hypothesis.

2. State the level of significance.

3. Identify the sampling distribution.

(The correlation is not significant)

(The correlation is significant)

= 0.01


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t0 4.0324.032

Rejection Regions

Critical Values t0

4. Find the critical value.

5. Find the rejection region.

6. Find the test statistic.


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t0

4.032 4.032

t= 9.811 falls in the rejection region. Reject the null hypothesis.

Thereisa significant correlation between the number oftimes absent and final grades.

7. Make your decision.

8. Interpret your decision.


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Linear Regression

Section 9.2


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The equation of a line may be written as y= mx+ bwhere mis the slope of the line and bis the y-intercept.

The line of regression is:

The slope mis:

The y-intercept is:

Once you know there is a significant linear correlation,you can write an equation describing the relationshipbetween thexandyvariables. This equation is called theline of regressionor least squares line.

The Line of Regression


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180

190

200

210

220

230240

250

260

1.5 2.0 2.5 3.0Ad $

= a residual

(xi,yi) = a data point

revenue

= a point on the line with the same x-value


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Calculate mand b.

Write the equation of theline of regression with

x= number of absencesand y= final grade.

The line of regression is: =

3.924x+ 105.667

60848464

81003364184954766561

624184

450696645666486

57 516 3751 579 39898

1 8 782 2 92

3 5 904 12 585 15 436 9 747 6 81

644

251442258136

xy x2 y2x y


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0 2 4 6 8 10 12 14 16

4045

50556065707580859095

Absences

FinalGra

de

m= 3.924 and b= 105.667

The line of regression is:

Note that the point = (8.143, 73.714) is on the line.

The Line of Regression


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The regression line can be used to predict values of yforvalues ofxfalling within the range of the data.

The regression equation for number of times absent and final grade is:

Use this equation to predict the expected grade for a student with

(a) 3 absences (b) 12 absences

(a)

(b)

Predicting yValues

=

3.924(3) + 105.667 = 93.895

= 3.924(12) + 105.667 = 58.579

= 3.924x+ 105.667


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Measures ofRegression andorrelation

Section 9.3


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The coefficient of determination, r2,is the ratio of explainedvariation in yto the total variation in y.

The correlation coefficient of number of times absent andfinal grade is r= 0.975. The coefficient of determinationis r2 = (0.975)2 = 0.9506.

Interpretation: About 95% of the variation in final gradescan be explained by the number of times a student isabsent. The other 5% is unexplained and can be due tosampling error or other variables such as intelligence,

amount of time studied, etc.

The Coefficient of Determination


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The Standard Error of Estimate,se,is the standard

deviation of the observed yivalues about the predicted

value.

The Standard Error of Estimate


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1 8 78 74.275 13.87562 2 92 97.819 33.86083 5 90 86.047 15.6262

4 12 58 58.579 0.33525 15 43 46.807 14.49326 9 74 70.351 13.31527 6 81 82.123 1.2611

92.767

= 4.307

x y

Calculate for eachx.

The Standard Error of Estimate


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Given a specific linear regression equation andx0, a specific valueofx, a c-prediction interval for yis:

where

Use a t-distribution with n 2 degrees of freedom.

The point estimate is andEis the maximum error of estimate.

Prediction Intervals


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Construct a 90% confidence interval for a final grade when astudent has been absent 6 times.

1. Find the point estimate:

The point (6, 82.123) is the point on the regression line withx-coordinate of 6.

Application


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Construct a 90% confidence interval for a final grade whena student has been absent 6 times.

2. Find E,

At the 90% level of confidence, the maximumerror of estimate is 9.438.

Application


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Construct a 90% confidence interval for a final gradewhen a student has been absent 6 times.

Whenx= 6, the 90% confidenceinterval is from 72.685 to 91.586.

3. Find the endpoints.

Application

E= 82.123

9.438 = 72.685

+ E= 82.123 + 9.438 = 91.561

72.685 < y< 91.561


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Regression Analysis

The regression equation is

y= 106

3.92x

Predictor Coef StDev T PConstant 105.668 3.655 28.91 0.000

Minitab Output

x 3.9241 0.4019 9.76 0.000

S = 4.307 R-Sq = 95.0% R-Sq(adj) = 94.0%


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Multiple Regression

Section 9.4


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Absence IQ Grade

More Explanatory Variables

8

2

5

12

15

9

6

115

135

126

110

105

120

125

78

92

90

58

43

74

81


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Regression Analysis


Grade = 52.7

2.65 absence + 0.357 IQ

Predictor Coef StDev T P

Constant

AbsenceIQ

Minitab Output

S = 4.603 R-Sq = 95.4% R-Sq(adj) = 93.2%

0.5730.277

0.571

0.611.26

0.62

86.1102.111

0.580

52.7202.652

0.357


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Interpretation


Grade = 52.7 2.65 absence + 0.357 IQ

When other variables are 0, the grade is 52.7.

If IQ is held constant, each time there is one moreabsence the predicted grade will decrease by 2.65points.

If number of absences is held constant, and IQ isincreased by one point the predicted grade will increaseby 0.357 points.


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The regression equation isGrade = 52.7 2.65 absence + 0.357 IQ

Predicting the Response Variable

Use the regression equation to predict a grade when astudent is absent 5 times and has an IQ of 125.

Grade = 52.7 2.65 absence + 0.357 IQ

Grade = 52.7 2.65(5) + 0.357(125) = 80.075 (about 80)

Use the regression equation to predict a grade when astudent is absent 9 times and has an IQ of 120.

Grade = 52.7 2.65 absence + 0.357 IQ

Grade = 52.7 2.65(9) + 0.357(120) = 71.69 (about 72)

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