Linear Correlation and

Regression Analysis

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SCATTER DIAGRAM

Positive Linear Correlation

Positive Correlation

Variables will tend to pair

as high measurements for

x with high measurements

for y and low x with low y

Negative correlation

Negative Correlation

Variables will tend to pair high x measurements with low measurements for y

and

low x with high y.

No correlation

THE COEFFICIENT OF LINEAR CORRELATION

Pearson’s Correlation Coefficient, r,

measures the strength of a linear

relationship between two variables for

a sample.

2222

yynxxn

yxxynr

Interpreting the Values of r r = 1

Perfect Positive Correlation

(Positive Slope)

Interpreting the Values of r r = -1

Perfect Negative correlation

(Negative Slope)

Interpreting the Values of r

r = 0

no correlation

Possible Values of r

–1 ≤ r ≤ 1

strong correlation weak correlation

r = -1 r = 0 r = +1

l

r = -0.84 r = -0.25 r = 0.07 r = 0.69

Calculate r

1-Var Stat TEMP1 and TEMP2

What is…

xy

TESTING THE SIGNIFICANCE OF THE CORRELATION COEFFICIENT

Is there a significant correlation between:

HS average and success in college?

Hours of T.V. watched and IQ score?

Age and blood pressure?

Advertising expenditures and sales?

Mothers shoe size and daughters IQ?

Cautions Regarding the Interpretation of Correlation Results

Caution #1:

Don’t Overlook the Possibility of

a Non-Linear Relationship

Two variables to have a linear correlation near zero, yet they could have a significant non-linear relationship.

Caution #2: Correlation Doesn’t Indicate a Cause-and-Effect Relationship

The number of storks nesting in

various European towns in the early 1900’s and the number of human babies born in the same towns had a very high correlation. However, we can’t conclude that an increase in the number of storks will cause an increase in the number of babies.

TESTING THE SIGNIFICANCE OF THE CORRELATION COEFFICIENT

The population correlation coefficient

is symbolized by Greek letter rho: ρ

The sample correlation coefficient r is

an estimate of the population

correlation coefficient.

Null Hypothesis Form

Ho: The population correlation

coefficient is equal to zero.

Ho: ρ = 0

There is no linear correlation between the two variables.

0 r

3 Forms of the Alternative Hypothesis

Ha:

Ha: ρ > 0

There is a positive correlation between the two variables.

Interpretation of Ha: Form #1

formular

General Solution (positive) 1TT >

0 rr

Ha: ρ < 0

There is a negative correlation between the two variables.

Interpretation of Ha: Form #2

formular0 r

General Solution (negative) 1TT >

r

Ha: ρ ≠ 0

There is a (some) correlation between the two variables.

Interpretation of Ha: Form #3

formular r0 r

General Solution (some) 2TT >

r

Degrees of freedom for

testing the correlation

coefficient

Subtract two from the

number of pairs of data

df = n – 2

Coefficient of Determination

is the influence that the

variance in the

independent variable

has on the

dependent variable.

is unexplained.

2r

21 r

Determine the variables

grades vs. study hours

independent is… dependent is…

Determine the variables

credits vs. age

sun light vs. height of a plant

bike vs. presidents

gift vs. work

LINEAR REGRESSION ANALYSIS

Regression Line Formula:

y' = a + bx

y' is the predicted value of y,

the dependent variable,

given

the value of x,

the independent variable.

Example

A scientists wants to

determine if there is a linear

relationship between the

amount of rainfall in May and

the number of mosquitoes. For

each of the selected years,

data pairs have listed in the

table.

Sample data

YEAR MOSQUITO

INDEX RAIN

2002 6.4 5.3

2001 2.3 2.7

2000 2.7 3.9

1999 1.2 2.4

1998 5.2 5.7

1997 2.6 3.5

Enter data

Scatter Plot

Procedure

a) Construct a scatter diagram.

b) Calculate the sample correlation coefficient, r.

c) Determine if r is significant at

α = 1%.

Procedure

If r is significant at α = 1%,

then continue…

Procedure d) Find and interpret its meaning.

e) Determine the regression equation, y'.

f) Using the regression equation, predict

the mosquito population index if we

have 3.1 inches of rain in May.

2r

CORRECT ORDER?

Perform Linear Reg T Test

Select Ha; Set Y1 (once)

Record : p-value, a, b,

Calculate, arrow down

randr2

Interpretations

r = 0.9339244305: close to r = 1

therefore a Strong Positive correlation.

p-value 0.006404729 is less then α = 1%,

so reject Ho and accept Ha.

There is a correlation between rainfall and the mosquito population index.

Coefficient of determination:

87.22% of the dependent

variable variation is attributed

to variation in the independent

variable;

12.78% is unexplained.

8722148419.02 r

Regression Equation

General form: y' = a + b (x)

y' = -1.9588137778595 + 1.368207773(x)

Prediction for the mosquito index:

Substitute rain = 3.1 inches

y' = 2.282630319

How much Confidence in the

Prediction of 2.282 population

index?

Since r = 0.9339244305, is very close to r = 1 (a perfect correlation),

there is a high level of confidence in the accuracy of the prediction.

Interpolation vs. Extrapolation

calculate this:

Use a rain prediction = 6739

Substitute Y1(6739)

Result =

How much confidence in the result ? NONE!

Using the Regression Line

BLOOD PRESSURE

Interpolate if maximum weight collected was

use any value within the

data max and min

256 lbs

BODY WEIGHT

Using the Regression Line

BLOOD PRESSURE

Interpolate Extrapolate

256 lbs 9842

BODY WEIGHT

Redo Scatter Plot for Regression line

Manually Calculate the correlation coefficient, r

for Rain & Mosquito Data

2222

yynxxn

yxxynr

EVtheisrcalculated 9339.0

SECOND METHOD

Use the table

to generate

the critical value:

r

Determine value from page 810 table V

Pairs of mosquito & rain values, n= 6

Subtract two from the number of

pairs of data values to generate

Degrees of freedom:

df = n – 2 = 4

We were given α = 1% 2 Tail

rcritical

9339.0rcalculated92.0rcritical

Example Craig, a marketing executive for a

microbrewery company, wants to determine

if there is a positive linear relationship

between advertising expenditures and sales

for their new lite beer product, Less Ale. He

randomly samples data for the past 7 sales

years and records the sample data in the

following table, where advertising

expenditure is measured in thousands and the beer sales in millions of dollars.

Expenditure Sales

Year (thousand $) (million $)

2003 35 38

2004 47 35

2005 65 42

2006 92 50

2007 55 40

2008 25 35

2009 82 48

a) Construct a scatter diagram.

b) Calculate the sample correlation

coefficient, r.

c) Determine if r is significant at a = 1%.

If r is significant at a = 1%, then do parts

d,e,f.

d) Find and interpret its meaning.

e) Determine the regression equation, y'.

f) Using the regression equation, predict the

beer sales for the advertising budget expenditure of $50,000.

2r

A prominent psychologist wonders if a patient’s score on the extrovert scale (those who seek out social environments) is positively correlated with time spent on social network sites online. She gathers data from 20 patients for both variables; the results are as follows:

Can the psychologist conclude that scores on an extroversion scale are significantly correlated with time spent on social network sites? Test at an alpha level of 1%.

a) Generate the scatter diagram for the data.

b) Calculate the sample correlation coefficient, r.

c) Determine if r is significant at a = 1%.

d) Determine the regression equation, y'.

Is there a positive correlation between

GIFT and AGE in class 3 at the 5% alpha level ?

Predict a GIFT for an AGE of 108. Do everything.

(only use the calculator )