Warm-upwith 3.3 Notes on Correlation

Universities use SAT scores in the admissions process because theybelieve these scores provide some insight into how a high schoolstudent will perform at the college level. Suppose the enteringfreshmen at a certain college have mean combined SAT scores of 1222with a standard deviation of 83. In the first semester these studentsattained a mean GPA of 2.66 with a standard deviation of 0.56. Ascatterplot showed the association to be reasonably linear, and thecorrelation between SAT score and GPA was 0.47.

a) Identify the following:

b) If r is the correlation calculate the slope b1.

c) Plug in and b1 to solve for b0 (y-intercept).

d) Write the equation of the regression line.

x

y

s

srb

xbyb

xbby

1

10

10ˆ

yx ssyx and,,,

yx,

Student of the day!Block 1

Student of the day!Block 2

Answers to A.P. Statistics H.W. 3.2 P#3 and 4, E # 9 and 11P4. a. The slope is about 0.8 b. This means as hand length increases

by 1 cm, the width would increase by 0.8 cm. c. d. A possible explanation is that the points represent those

students that measured hand width by not spreading fingers apart. If the cluster of points below was removed then the slope

becomes steeper.P5.

lengthhand8.07.1widthhand

P#5 f and g.

E #9

E #10

Calculating the LSRL step-by-step

A high school counselor wants to see if there is acorrelation between GPA and SAT score. Here are some ofthe data the guidance counselor is looking at.

Sum: _____ _____ _____ ______ _______ _______Mean: ____ _____ _____ _____ to find the bo substitute ( )

xbby o 1ˆ

yx,

Use the calculator to find the LSRL

STAT = > CALC Select #8 LinReg(a + bx)

To get correlation go to 2nd 0 for Catalog hit x-1for D.Find DiagonsticOn highlight it and hit ENTER two

times.

3.3 CorrelationCorrelation tells you the strength of the trend.

OR

r2 is the coefficient of determination. It is the proportion of variation of the response variable that can be explained by the explanatory variable.

1

1i i

x y

x x y yrn s s

1

( )( )1 x yr z z

n

How to see the residual plot on the calculator!

If the regression line is a good model, we would expect tofind the residuals more or less randomly scattered aboutthe average residual which is represented by thehorizontal line. No pattern on the residual plot means thedata could be represented by a linear model.

Finding Correlation Coefficient on the Calculator• In order to find r (correlation coefficient) on your calculator. • We need to change it from the factory setting by going to

CATALOG (Hit 2nd the 0 zero). • When in CATALOG your keys are automatically in Alpha so

hit x-1 (where the Green D is located)• Scroll down to DiagnosticOn and press Enter twice!

Practice: Speed: 20 30 40 50 60 MPG: 24 28 30 28 24

STAT -> Calc Scroll down to 8: LinReg(a+bx) EnterLineReg(a +bx) L1, L2, Y1 Enter

Notice you get both r and r2!Any theories about the r in this problem? Look at the graph.

Let’s look at the residual plot

3.3 Pg 141 P#15 and 16, E#27 and 28

Investigation ofScatter plots and Residuals

Any observations on the two Data Sets?

Remember to use

Zoom #9 Zoom Stat Each time.

Friday is 3.1 to 3.3 Quiz• Know all vocabulary and formulas from 3.1 to 3.3• Know how create a scatter plot and find the Line of best fit.• Know how to calculate the LSRL, how to extrapolate and

calculate the residual for a particular piece of data.• Be able to interpret the slope and y-intercept in the context

of the problem• Know how to describe scatter plots: Shape, Trend Strength

and under Shape: Linearity, Clusters and outlier.• Know how to calculate the SSE