copyright © 2010, 2007, 2004 pearson education, inc. all rights reserved. 10.1 - 1 chapter 10...

Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 1

Chapter 10Correlation and Regression

10-2 Correlation (Part 1 Only)

10-3 Regression (Part 1 Only)


Preview

In this chapter we introduce methods for determining whether a correlation, or association, between two variables exists and whether the correlation is linear. For linear correlations, we can identify an equation that best fits the data and we can use that equation to predict the value of one variable given the value of the other variable.


Section 10-2 Correlation


Key Concept

Part 1 of this section introduces the linear correlation coefficient r, which is a numerical measure of the strength of the relationship between two variables representing quantitative data.

Using paired sample data (sometimes called bivariate data), we find the value of r (usually using technology), then we use that value to conclude that there is (or is not) a linear correlation between the two variables.


Key Concept

In this section we consider only linear relationships, which means that when graphed, the points approximate a straight-line pattern.


Part 1: Basic Concepts of Correlation


Definition

A correlation exists between two variables when the values of one are somehow associated with the values of the other in some way.


Definition

The linear correlation coefficient r measures the strength of the linear relationship between the paired quantitative x- and y-values in a sample.


Exploring the Data

We can often see a relationship between two variables by constructing a scatterplot.

Figure 10-2 following shows scatterplots with different correlation coefficients.


Scatterplots of Paired Data

Figure 10-2



Figure 10-2


• Example: problem 10 on page 509

x y

10 7.46

8 6.77

13 12.74

9 7.11

11 7.81

14 8.84

6 6.08

4 5.39

12 8.15

7 6.42

5 5.73

(a) Construct a scatterplot


Scatterplot with Graphing Calculator

• Enter the X data values in two lists.

• Press 2nd STATPLOT and choose #1 PLOT 1. Be sure the plot is ON, the scatter plot icon is highlighted (top row, first icon) •Enter the list of the X data values next to Xlist, and the list of the Y data values next to Ylist. •Choose any of the three marks. Press ZOOM and #9 ZoomStat.



(a) Scatterplot

Note the outlier at x=13


Properties of the Linear Correlation Coefficient r1. –1 r 1

2. if all values of either variable are converted to a different scale, the value of r does not change.

3. the value of r is not affected by the choice of x and y. Interchange all x- and y-values and the value of r will not change.

4. r measures strength of a linear relationship (the closer r is to positive or negative 1, the “more linear” the relationship)


Formula 10-1

nxy – (x)(y)

n(x2) – (x)2 n(y2) – (y)2r =

The linear correlation coefficient r measures the strength of a linear relationship between the paired values in a sample.

Computer software or calculators can compute r

Formula For r


Notation for the Linear Correlation Coefficient

n = number of pairs of sample data

denotes the addition of the items indicated.

x denotes the sum of all x-values.

x2 indicates that each x-value should be squared and then those squares added.

(x)2 indicates that the x-values should be added and then the total squared.


Notation for the Linear Correlation Coefficient

xy indicates that each x-value should be first multiplied by its corresponding y-value. After obtaining all such products, find their sum.

r = linear correlation coefficient for sample data.

= linear correlation coefficient for population data.



x y

10 7.46

8 6.77

13 12.74

9 7.11

11 7.81

14 8.84

6 6.08

4 5.39

12 8.15

7 6.42

5 5.73

(b) Find the linear correlation coefficient r and determine if there is sufficient evidence to support the claim of a linear correlation between x and y.


(see page 508 of your book)

• Enter the data in two lists.

• Press STAT and select TESTS

• LinRegTTest is option F (scroll arrow up 3 places)

• Enter the names of the lists from step 1.

• Arrow down to Calculate and then press Enter

• The r value is the last value displayed; round this value to three decimal places

Calculate r Using Calculator



Calculate r Using Calculator

x y

10 7.46

8 6.77

13 12.74

9 7.11

11 7.81

14 8.84

6 6.08

4 5.39

12 8.15

7 6.42

5 5.73

(b) Using calculator we find that r = 0.816


Interpreting r

Using Table A-5:

• Find the row with same value of n as the data set

• If the absolute value of the computed value of r, denoted |r|, is greater than the number in the first column in Table A-5, conclude that there is a linear correlation. Otherwise, there is not sufficient evidence to support the conclusion of a linear correlation.


Table A-5

(b) For n=11 critical value of r is 0.602



Interpret r

x y

10 7.46

8 6.77

13 12.74

9 7.11

11 7.81

14 8.84

6 6.08

4 5.39

12 8.15

7 6.42

5 5.73

(b) Comparing 0.602 with r = 0.816, there is a (positive) linear correlation since r is greater than 0.602



Interpret r

(c) Identify the feature of the data that would be missed if part (b) was completed without the use of a scatterplot

ANSWER:

scatterplot shows the relationship is linear except for one point (an error or an outlier at x=13)


Properties of the Linear Correlation Coefficient r5. r is very sensitive to outliers, they can

dramatically affect its value.

If (13,12.74) is removed from the list in the previous example, the calculator gives

r = 0.999996554


NOTE: if r is negative, take its absolute value before comparing with value in Table A-5

• Example: r = -0.375 and n=17

Interpret r

From Table A-5, for n=17 we get a value of 0.482

Comparing |r|=0.375 there is not evidence for a linear correlation since |r| is less than 0.482


Caution

Know that the methods of this section apply to a linear correlation. If you conclude that there does not appear to be linear correlation, know that it is possible that there might be some other association that is not linear.


Interpreting r:Explained Variation

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


Using example problem 10 on page 509, the linear correlation coefficient was r = 0.816. The proportion of y can be explained by the variation in x:

r2 =(0.816)(0.816)=0.666

We conclude that 0.666 (or about 67%) of the variation in y can be explained by the variation in x. This implies that about 33% of the variation in y cannot be explained by the variation in x.

Example:


Common Errors Involving Correlation

1. Causation: It is wrong to conclude that correlation implies causality.

2. Averages: Averages suppress individual variation and may inflate the correlation coefficient.

3. Linearity: There may be some relationship between x and y even when there is no linear correlation.


Recap

In this section, we have discussed:

Correlation.

The linear correlation coefficient r.

Requirements, notation and formula for r.

Interpreting r.


Section 10-3 Regression


Key Concept

In part 1 of this section we find the equation of the straight line that best fits the paired sample data. That equation algebraically describes the relationship between two variables.

The best-fitting straight line is called a regression line and its equation is called the regression equation.


Part 1: Basic Concepts of Regression


Regression

The regression equation expresses a relationship between x (called the explanatory variable, predictor variable or independent variable), and y (called the response variable or dependent variable).

^


Definitions

Regression EquationGiven a collection of paired data, the regression equation

Regression Line

The graph of the regression equation is called the regression line (or line of best fit, or least squares line).

y = b0 + b1x^

algebraically describes the relationship between the two variables.


The regression line is the best linear fit of the sample points.

Special Property


Regression

The typical equation of a straight line:

m = slope of the line (coefficient of x)b = y-intercept of the line (constant)

y = m x + b


Notation for Regression Equation (Sample)

y-intercept of regression equation

Slope of regression equation

Equation of the regression line

SampleStatistic

b0

b1

y = b0 + b1x

^


Notation for Regression Equation (Population)

y-intercept of regression equation

Slope of regression equation

Equation of the regression line

PopulationParameter

0

1

y = 0 + 1 x


Requirements

1. The sample of paired (x, y) data is a random sample of quantitative data.

2. Visual examination of the scatterplot shows that the points approximate a straight-line pattern.

3. Any outliers must be removed if they are known to be errors. Consider the effects of any outliers that are not known errors.


Formulas for b0 and b1

Formula 10-3 (slope)

(y-intercept)Formula 10-4

calculators or computers can compute these values

(round to three significant digits)

b0y b

1x

b

1r

sy

sx


Calculator

1. Enter the data in two lists.

2. Make a scatter plot of the data (use 2nd Y= to get STAT PLOT, choose Plot1 On, first scatterplot icon, then zoom 9)

(your plot will look different)


Calculator

3. Plot the regression line. Choose:

4.

STAT → CALC #4 LinReg(ax+b)

Include the parameters L1, L2, Y1.

NOTE: Y1 comes from VARS → YVARS, #Function, Y1


Calculator

5. Choose Y= and the equation for the regression line will be stored in Y1

Then choose GRAPH and the regression line will be plotted.


Calculator

6. Choose TRACE and you can see X and Y values on scatterplot or regression line



Calculate Regression Line

x y

10 7.46

8 6.77

13 12.74

9 7.11

11 7.81

14 8.84

6 6.08

4 5.39

12 8.15

7 6.42

5 5.73



Scatterplot




Regression line


xy 500.000.3ˆ


Calculator

A quick method of determining linear regression coefficients (without graphing the line)

Choose:

Enter lists and then Calculate

You can then graph the line by entering equation by hand in Y=

STAT → TESTS F LinRegTTest


1. Use the regression equation for predictions only if the graph of the regression line on the scatterplot confirms that the regression line fits the points reasonably well.

Using the Regression Equation for Predictions

2. Use the regression equation for predictions only if the linear correlation coefficient r indicates that there is a linear correlation between the two variables (as described in Section 10-2).


3. Use the regression line for predictions only if the data do not go much beyond the scope of the available sample data. (Predicting too far beyond the scope of the available sample data is called extrapolation, and it could result in bad predictions.)

Using the Regression Equation for Predictions

4. If the regression equation does not appear to be useful for making predictions, the best predicted value of a variable is its point estimate, which is its sample mean.


Strategy for Predicting Values of Y


• Use results from problem 10 on page 526 predict y when x=6.25

Regression line is:

The r value for the data was 0.816 (see example on previous slides from 10-2) which is greater than critical r value from Table A-5, so there is a linear correlation

xy 500.000.3ˆ

Calculate y From Regression Line


Predicted y value is:

125.6)25.6(500.000.3ˆ y



• Problem 6 on page 525

Given n=8 (“eight pairs”) and r = 0.693

Using Table A-5 for n=8 gives critical r value as 0.707. Since r is less than this value, there is not a linear correlation in the data. Use the given mean of the daughter’s heights to estimate y.


inches 3.63ˆ yy

copyright © 2010, 2007, 2004 pearson education, inc. all rights reserved. 10.1 - 1 chapter 10...

Documents