Essential Statistics Chapter 5 1

Least Squares Regression Line

Regression line equation: y = a + bx^

– x is the value of the explanatory variable– “y-hat” is the predicted value for a x value

– a and b are just the intercept and slope of a straight line

Essential Statistics Chapter 5 2

^

xbya

s

srb

x

y

Regression equation: y = a + bx

Regression Line Calculation

where sx and sy are the standard deviations of the two variables, and r is their correlation

R-square (R2) To assess how well regression equation (a

model) explain and predicts future outcomes R-square value (0 ≤ r2 ≤ 1) In model analysis, measuring the accuracy of

regression equation

◙ value of 1 indicates a reliable model for

future prediction ◙ value of zero indicates the model fails.

Essential Statistics Chapter 5 3

Essential Statistics Chapter 5 4

Coefficient of Determination (R2) In linear regression, R2 is the square of the

sample correlation coefficient R2 or r2 is the fraction of the explained variation to

the total variation in the values of the response variable (y)

◙ r=1: r2=1: regression line explains all (100%) ofthe variation in y

◙ r=.7: r2=.49: regression line explains almost half

(50%) of the variation in y

Essential Statistics Chapter 5 5

Residuals

A residual is the difference between an observed value of the response variable and the value predicted by the regression line:

residual = y y

Essential Statistics Chapter 4 6

Measuring Strength & Directionof a Linear Relationship

The correlation coefficient r– measure of the strength of the relationship:

the stronger the relationship, the larger the magnitude of r.

– measure of the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship.

Essential Statistics Chapter 1 7

Weight Data: Histogram

0

2

4

6

8

10

12

14

Frequency

100 120 140 160 180 200 220 240 260 280Weight

* Left endpoint is included in the group, right endpoint is not.

Nu

mb

er

of s

tude

nts

Essential Statistics Chapter 1 8

Weight Data:Stemplot

(Stem & Leaf Plot)

10 016611 00912 003457813 0035914 0815 0025716 55517 00025518 00005556719 24520 321 02522 023242526 0

Key

20|3 means203 pounds

Stems = 10’sLeaves = 1’s

Essential Statistics Chapter 2 9

Five-Number Summary

minimum = 100 Q1 = 127.5 M = 165 Q3 = 185 maximum = 260

The middle 50% of the data are between Q1 and Q3

Essential Statistics Chapter 2 10

M

Weight Data: Boxplot

Q1 Q3min max

100 125 150 175 200 225 250 275

Weight