Section 10.4-1Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Lecture Slides

Elementary Statistics Twelfth Edition

and the Triola Statistics Series

by Mario F. Triola

Section 10.4-2Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Chapter 10Correlation and Regression

10-1 Review and Preview

10-2 Correlation

10-3 Regression

10-4 Prediction Intervals and Variation10-5 Multiple Regression

10-6 Nonlinear Regression

Section 10.4-3Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Key ConceptIn this section we present a method for constructing a prediction interval, which is an interval estimate of a predicted value of y.

(Interval estimates of parameters are confidence intervals, but interval estimates of variables are called prediction intervals.)

Section 10.4-4Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Requirements

For each fixed value of x, the corresponding sample values of y are normally distributed about the regression line, with the same variance.

Section 10.4-5Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Formulas

For a fixed and known x0, the prediction interval for an individual y value is:

with margin of error:

ˆ ˆy E y y E

20

/2 22

11e

n x xE t s

n n x x

Section 10.4-6Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Formulas

The standard error estimate is:

(It is suggested to use technology to get prediction intervals.)

Section 10.4-7Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

If we use the 40 pairs of shoe lengths and heights, construct a 95% prediction interval for the height, given that the shoe print length is 29.0 cm.

Recall (found using technology, data in Appendix B):

0

/2

ˆ 80.9 3.2229.05.943761

ˆ 174.32.024 (Table A-3, df = 38, 0.05 area in two tails)

e

y xxsy

t

Section 10.4-8Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

The 95% prediction interval is 162 cm < y < 186 cm.

This is a large range of values, so the single shoe print doesn’t give us very good information about a someone’s height.

Section 10.4-9Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Explained and Unexplained VariationAssume the following:

•There is sufficient evidence of a

linear correlation.

•The equation of the line is

ŷ = 3 + 2x

•The mean of the y-values is 9.

•One of the pairs of sample data

is x = 5 and y = 19.

•The point (5,13) is on the

regression line.

Section 10.4-10Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Explained and Unexplained Variation

The figure shows (5,13) lies on the regression line, but (5,19) does not.

We arrive at:

Total Deviation (from 9) of the point (5, 19) 19 9 10.ˆExplained Deviation (from 9) of the point (5, 19) 13 9 4.

ˆUnexplained Deviation (from 9) of the point (5, 19) 19 13 6.

y y yy y yy y y

Section 10.4-11Copyright © 2014, 2012, 2010 Pearson Education, Inc.

(total deviation) = (explained deviation) + (unexplained deviation)

= +

(total variation) = (explained variation) + (unexplained variation)

Relationships

ˆ( )y y( )y y ˆ( )y y

2( )y y 2ˆ( )y y 2ˆ( )y y = +

Section 10.4-12Copyright © 2014, 2012, 2010 Pearson Education, Inc.

DefinitionThe coefficient of determination is the amount of the variation in y that is explained by the regression line.

The value of r2 is the proportion of the variation in y that is explained by the linear relationship between x and y.

2 explained variationtotal variation

r

Section 10.4-13Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

If we use the 40 paired shoe lengths and heights from the data in Appendix B, we find the linear correlation coefficient to be r = 0.813.

Then, the coefficient of determination is

r2 = 0.8132 = 0.661

We conclude that 66.1% of the total variation in height can be explained by shoe print length, and the other 33.9% cannot be explained by shoe print length.