HealeyCh15b

8/12/2019 HealeyCh15b

1/23

Chapter 15

Association Between VariablesMeasured at the Interval-Ratio

Level


2/23

Chapter Outline

Interpreting the CorrelationCoeicient! r "

#he Correlation Matri$ #esting %earson&s r or 'igniicance

Interpreting 'tatistics! #he Correlates

o Cri(e


3/23

'cattergra(s

'cattergra(s have two di(ensions! #he ) *independent+ variable is arra,ed

along the horiontal a$is.

#he / *dependent+ variable is arra,edalong the vertical a$is.


4/23

'cattergra(s

0ach dot on a scattergra( is a case.

#he dot is placed at the intersection

o the case&s scores on ) and /.


5/23

'cattergra (s

Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

'hows the relationship between College 0ducated *)+ and Voter #urnout*/+ on election da, or the 52 states.


6/23

'cattergra(s


43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

3oriontal ) a$is - o population o astate with a college education. 'cores range ro( 15.4 to 4.6

and increase ro( let to right.


7/23

'cattergra(s


43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

Vertical */+ a$is is voter turnout. 'cores range ro( .1 to 72. and

increase ro( botto( to top


8/23

'cattergra(s! Regression Line


43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

A single straight line that co(es as close aspossible to all data points.

Indicates strength and direction o therelationship.


9/23

'cattergra(s!'trength o Regression Line

#he greater the e$tent to which dots are clusteredaround the regression line8 the stronger therelationship.

#his relationship is wea9 to (oderate in strength.


43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

%College


10/23

'cattergra(s!:irection o Regression Line

%ositive! regression line rises let to right. ;egative! regression line alls let to right. #his a positive relationship! As college

educated increases8 turnout increases.Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

%College


11/23

'cattergra(s

Inspection o the scattergra( shouldalwa,s be the irst step in assessing thecorrelation between two I-R variables


43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College


12/23

#he Regression Line! score on the dependent variable a > the / intercept or the point where the

regression line crosses the / a$is.

b > the slope o the regression line or thea(ount o change produced in / b, a unitchange in )

) > score on the independent variable


13/23


14/23

Regression Anal,sis

#he / intercept *a+ is co(puted ro(


15/23

Regression Anal,sis

." a */ intercept+> 52.24

Regression or(ula! / > 52.24 ? ." ) A slope o ." (eans that turnout increases

b, ." *less than hal a percent+ or ever,unit increase o 1 in college educated.

#he / intercept (eans that the regressionline crosses the / a$is at / > 52.24.


16/23

%redicting /

=hat turnout would be e$pected in a statewhere onl, 12 o the population wascollege educated@

=hat turnout would be e$pected in a statewhere 72 o the population was collegeeducated@

#his is a positive relationship so the valueor / increases as ) increases! 128 / > 52.4 ?."*12+ > 5.5 728 / > 52.4 ? ."*72+ > 7.7


17/23

%earson correlation coeicient

But o course8 this is ust an esti(ate oturnout based on college educated8 and(an, other actors also aect voter

turnout. 3ow (uch o the variation in voter turnout

depends on college educated@ #herelevant statististic is the coeicient o

deter(ination *r suared+8 but irst weneed to learn about %earson&s correlationcoeicient *r+.


18/23

%earson&s r

%earson&s r is a (easure o association or I-Rvariables.

It varies ro( -1.2 to ?1.2 Relationship (a, be positive *as ) increases8 /

increases+ or negative *as ) increases8 / decreases+ .4". #he relationship is positive! as level o education

increases8 turnout increases.

3ow strong is the relationship@


19/23

0$a(ple o Co(putation

#he co(putation and interpretation o a8 b8and %earson&s r will be illustrated using%roble( 15.1.

#he variables are! Voter turnout */+

Average ,ears o school *)+

#he sa(ple is 5 cities. #his is onl, to si(pli, co(putations8 5 is (uch

too s(all a sa(ple or serious research.


20/23


#he scores on eachvariable aredispla,ed in table

or(at! / > #urnout

) > /ears o0ducation

Cit, ) /

A 11. 55

B 1".1 62

C 1".7 65

: 1".D 6D

0 14.2 72


21/23


Sums areneeded toco(pute b8 a8and %earson&s

r.

) / )"/"

)/

11. 55 11.61 42"5 65.5

1".1 62 16.1 4622 7"6

1".7 65 161." ""5 D"5.5

1".D 6D 164.D 6" D72.

14.2 72 16 22 12

62.5 318 782.15 20374 3986.4


22/23

Interpreting %earson&s r

An r o 2.D indicates an e$tre(el, strongrelationship between average ,ears oeducation and voter turnout or these ive

cities. #he coeicient o deter(ination is r"> .6.

Enowing education level i(proves ourprediction o voter turnout b, 6. #his is

a %R0 (easure *li9e la(bda and ga((a+ =e could also sa, that education e$plains

6 o the variation in voter turnout.


23/23

Interpreting %earson&s r

Our irst e$a(ple provides a (orerealistic value or r. #he r between turnout and college

educated or the 52 states was! r > .4" #his is a wea9 to (oderate8 positive

relationship.

#he value o r"

is .12.%ercent college educated e$plains12 o the variation in turnout.

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HealeyCh15b