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Chapter 3: Examining Relationships

3.1 Scatterplots

3.2 Correlation

3.3 Least-Squares Regression

y = 3.9951x + 4.5711

R2 = 0.9454

181920212223242526

3.5 4.0 4.5 5.0

Fiber Tenacity, g/den

Fabr

ic Te

nacit

y, lb/

oz/yd

^2

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Relationship Between Fiber Tenacityand Fabric Tenacity

Fiber Tenacity,g/den

Fabric Tenacity,lb/oz/yd2

3.6 19.0

3.9 20.5

4.1 20.8

4.3 21.0

4.8 23.0

5.0 24.9

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Variable Designations

• Which variable is the dependent variable?

– Our text uses the term response variable.

• Which variable is the independent variable?

– Explanatory variable

• Problems 3.1 and 3.4, p. 123

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Scatterplot 1: Relationship Between FiberTenacity and Fabric Tenacity

181920212223242526

3.5 4.0 4.5 5.0

Fiber Tenacity, g/den

Fab

ric

Ten

acit

y, lb

/oz/

yd^

2

Note placement of response and explanatory variables. Also noteaxes labels and plot title.

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Problem 3.6, p. 125

• Type data into your calculator.

• Examining a scatterplot:

– Look for the overall pattern and striking deviations from that pattern.

• Pay particular attention to outliers

– Look at form, direction, and strength of the relationship.

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Examining a Scatterplot, cont.

• Form

– Does the relationship appear to be linear?

• Direction

– Positively or negatively associated?

• Strength of Relationship

– How closely do the points follow a clear form?

– In the next section, we will discuss the correlation coefficient as a numerical measure of strength of relationship.

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Scatterplot for 3.6

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Problem 3.9, p. 129

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Tips for Drawing Scatterplots

• p. 128

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0

10

20

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60 70 80 90 100 110

Year (67=year 1967)

Inco

me

(Th

ou

san

ds

of

Yea

r 20

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Black Hispanic White Asian

Adding a Categorical Variable to a Scatterplot

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Homework

• Reading: pp. 121-135

• Problems:

– 3.11 (p. 129)

– 3.12 (p. 132) … on Excel

– 3.16 (p. 136)

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Which shows the strongest

relationship?

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30 40 50 60

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The two plots represent the same data!

• Our eye is not good enough in describing strength of relationship.

– We need a method for quantifying the relationship between two variables.

• The most common measure of relationship is the Pearson Product Moment correlation coefficient.

– We generally just say “correlation coefficient.”

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Correlation Coefficient, r

• The correlation, r, is an average of the products of the standardized x-values and the standardized y-values for each pair.

y

in

i x

i

s

yy

s

xx

nr

11

1

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Correlation Coefficient, r

• A correlation coefficient measures these characteristics of

the linear relationship between two variables, x and y.

– Direction of the relationship

• Positive or negative

– Degree of the relationship: How well do the data fit the

linear form being considered?

• Correlation of (1 or -1) represents a perfect fit.

• Correlation of (0) indicates no relationship.

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Interpreting Correlation Coefficient, r

• Correlation Applet: http://www.duxbury.com/authors/mcclellandg/tiein/johnson/correlation.htm

• Facts about correlation

– pp.143-144

• Correlation is not a complete description of two-variable data. We also need to report a complete numerical summary (means and standard deviations, 5-number summary) of both x and y.

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Exercise

• 3.25, p. 146

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Figure 3.5, p. 135

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Figure 3.6, p. 136

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Outlier, or influential point?

• Let’s enter the data into our calculators and calculate the correlation coefficient. The data are in the middle two columns of Table 1.10, p. 59.

– r=?

• Now, remove the possible influential point. What happens to r?

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Exercises: Understanding Correlation

• Review “Facts about correlation,” pp. 143-144

• 3.34, 3.35, and 3.37, p. 149

• Reading: pp. 149-157

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Relationship Between Winding Tensionand Yarn Elongation

y = -0.0759x + 9.4455

R2 = 0.732

6.0

6.5

7.0

7.5

8.0

8.5

9.0

10 15 20 25 30 35

Winding Tension, g

Elongation%

24(e)error yyresidual^

i

Least Squares Regression

• Ultimately, we would like to predict elongation by using a

more practical measurement, winding tension.

– A regression line, also called a line of best fit, was found.

• How was the line of best fit determined?

– Determine mathematically the distance between the line

and each data point for all values of x.

– The distance between the predicted value and the actual

(y) value is called a residual (or error).

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n

1i

2^

i2 )y(ye

• The best-fitting line is the line that has the smallest sum of e2 ... the least squares regression line! That is, the line of best fit occurs when:

minimum )y(yen

1i

2^

i2

Least Squares Regression: Line of Best Fit

• This could be done for each data point. If we square each residual and sum all of the squared residuals, we have:

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A Residual (Figure 3.11, p. 151)

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bxa ^

y

Least-Squares Regression Line

• With the help of algebra and a little calculus, it can be

shown that this occurs when:

x

y

s

srb

xbya

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Exercise 3.12, p. 132

• Is there a relationship between lean body mass and resting metabolic rate for females?

– Quantify this relationship.

• Find the line of best fit (the least-squares regression, LSR).

• Use the LSR to predict the resting metabolic rate for a woman with mass of 45 kg and for a woman with mass of 59.5 kg.

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Interpreting the Regression Model

• The slope of the regression line is important for the interpretation of the data:

– The slope is the rate of change of the response variable with a one unit change in the explanatory variable.

• The intercept is the value of y-predicted when x=0. It is statistically meaningful only when x can actually take values close to zero.

30r = 0.85, r2 = 0.72

1- r2 = 0.28

R2: Coefficient of Determination

• Proportion of variability in one variable that can be

associated with (or predicted by) the variability of the

other variable.

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Exercise 3.45, p. 166

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Exercise 3.45, p. 166

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Back to residuals …

• In regression, we see deviations by looking at the scatter of points about the regression line. The vertical distances from the points to the least-squares regression line are as small as possible, in the sense that they have the smallest possible sum of squares.

• Because they represent “left-over” variation in the response after fitting the regression line, these distances are called residuals.

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Examining the Residuals

• The residuals show how far the data fall from our regression line, so examining the residuals helps us to assess how well the line describes the data.

– Residuals Plot

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Residuals Plot

• Let’s construct a residuals plot, that is, a plot of the explanatory variable vs. the residuals.

– pp. 174-175

• The residuals plot helps us to assess the fit of the least squares regression line.

– We are looking for similar spread about the line y=0 (why?) for all levels of the explanatory variable.

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Residuals Plot Interpretation, cont.

• A curved or other definitive pattern shows an underlying relationship that is not linear.

– Figure 3.19(b), p. 170

• Increasing or decreasing spread about the line as x increases indicates that prediction of y will be less accurate for smaller or larger x.

– Figure 3.19(c), p. 171

• Look for outliers!

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Figures 3.19 (a-c), pp. 170-171

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How to create a residuals plot• Create regression model using your calculator.

• Create a column in your STAT menu for residuals. Remember that a residual is the actual value minus the predicted value:

yyresidual

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Residuals Plot for 3.45

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HW

• Read through end of chapter

• Problems:

– 3.42 and 3.43, p. 165

– 3.46, p. 173

• Chapter 3 Test on Friday

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Regression Outliers and Influential Observations

• A regression outlier is an observation that lies outside the overall pattern of the other observations.

• An observation is influential for a statistical calculation if removing it would markedly change the result of the calculation.– Points that are outliers in the x direction of a scatterplot are

often influential for the least-squares regression line.• Sometimes, however, the point is not influential when it falls in

line with the remaining data points.

– Note: An influential point may be an outlier in terms of x, but we label it as “influential” if removing it significantly influences the regression.

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Practice Problems

• Problems:

– 3.56, p. 179

– 3.74, p. 188

– 3.76, p. 189

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Preparing for the Test

• Re-read chapter.

– Know the terms, big concepts.

• Chapter Review, pp. 181-182

• Go back over example and HW problems.

• Study slides!