Chapter 3 -...

Chapter 3

Examining Relationships

Lesson 3-1

Scatterplots

Variables

Response variable (dependent)

Measures the outcome of a study.

Explanatory variable (independent)

Attempts to explain the observed outcomes

Example – Page 123, #3.4

The most common treatment for breast cancer was once

the removal of the breast. It is now usual to remove only

the tumor and nearby lymph nodes, followed by radiation.

The change in policy was due to a large medical experiment

that compared two treatments. Some breast cancer patients,

chosen at random, were given each treatment. What

are the explanatory and response variable? Are they

categorical or quantitative?

Explanatory variable Response variable

Treatment old or new

Categorical

Survival Time

Quantitative

Scatterplot

A scatterplot shows the relationship between two

quantitative variables measured on the same individuals.

Always plot the explanatory variable (if there is one) on the

horizontal axis (x-axis) of a scatterplot. The response

variable on the vertical axis (y-axis).

If there is no explanatory-response distinction, either

variable can go on the horizontal axis.


Manatees are large, gentle sea creatures that live

along the Florida coast. Many manatees are killed

or injured by powerboats. Here are data on powerboat

registrations (in thousands) and the number of manatees

killed by boats in Florida in the years 1977 to 1990:

A. We want to examine the relationship between number

of power boats and number of manatee killed by

boats. Which is the explanatory variable

Example – Page 125, #3.6Manatees Killed by Boats in Florida from 1977 to 1990

Year Powerboat

(Thousands)

Manatees

killed

Year Powerboat

(Thousands)

Manatees

Killed

1977 447 13 1984 559 34

1978 460 21 1985 585 33

1979 481 24 1986 614 33

1980 498 16 1987 645 39

1981 513 24 1988 675 43

1982 512 20 1989 711 50

1983 526 15 1990 719 47


A. We want to examine the relationship between number

of power boats and number of manatee killed by

boats. Which is the explanatory variable

Explanatory Variable = Number of powerboat registrations

B. Make a scatterplot of these data. (Be sure to label the

axes with variable names, not just x and y.) What does

the scatterplot show about the relationship between these

variables?

Example – Page 125, #3.6Manatees Killed by Boats in Florida from 1977 to 1990

Year Powerboat

(Thousands)

Manatees

killed

Year Powerboat

(Thousands)

Manatees

Killed

1977 447 13 1984 559 34

1978 460 21 1985 585 33

1979 481 24 1986 614 33

1980 498 16 1987 645 39

1981 513 24 1988 675 43

1982 512 20 1989 711 50

1983 526 15 1990 719 47


2nd Y=

ZOOMSTAT


0

10

20

30

40

50

60

400 450 500 550 600 650 700

Boats (Thousands)

Man

ate

es K

ille

d

The plot shows a moderately strong linear relationship.

As registrations increase, the number of manatee deaths

also tends to increase.

Describing A Scatterplot Direction (think slope)

Positive Direction

Negative Direction

Form Linear Relationship

Curved Relationship

Clusters

Scatter How closely the points follow a clear form.

Outliers An individual values that falls outside the overall

pattern of the relationship

Example – Page 129, #3.9Use the scatterplot from Exercise 3.6 (Page 125). You

make a scatterplot of power boats registered in Florida

and manatees killed by boats.

A. Describe the direction of the relationship. Are the

variables positively or negatively associated?

B. Describe the form of the association. Is it linear?

C. Describe the strength of the relationship. Can the number

of manatees killed be predicted accurately from powerboat

registrations? If powerboat registrations remained constant

at 719,000, about how many manatees would be killed by

boats each year?


A. Describe the direction of the relationship. Are the

variables positively or negatively associated?

The variables are positively associated; that is, at the

number of jet skis in use increases, the number of

manatees killed also increase.

B. Describe the form of the association. Is it linear?

The association is moderately linear.


C. Describe the strength of the relationship. Can the number

of manatees killed be predicted accurately from powerboat

registrations? If powerboat registrations remained constant

at 719,000, about how many manatees would be killed by

boats each year?

The association is relatively strong. The number of manatees

killed can be predicted accurately from the number of

powerboat registrations. If the number of registrations remains

constant 719,000, we would expect between 45 and 50

manatees to be killed per year.


Calories and hot dogs that are high in calories are also high

in salt? The following slide is a scatterplot of the calories

and salt content (measured as milligrams of sodium) in

17 brands of meat hot dogs.


A. Roughly what are the lowest and highest calories counts

among these brands? Roughly what is the sodium level in

the brands with the fewest and with the most calories.

B. Does the scatterplot show a clear positive or negative

association? Say in words what this association means

about calories and salt in hot dogs.

C. Are there any outliers? is the relationship (ignoring outliers)

roughly linear in form? Still ignoring outliers, how strong

would say the relationship between calories and sodium is?

Example

Example – Page 136, 3.16

A. Lowest: about 107 calories with about 145 mg of sodium

Highest: about 195 calories with about 510 mg of sodium

B. There is positive association: high calorie hot dogs tend

to be high in salt, and low calorie hot dogs tend to have

low sodium.

C. The lower left point is an outlier. Ignoring the point,

the remaining points seem to fall roughly on a line.

The relationship is moderately strong.

Lesson 3-2

Correlation

Correlation

Correlation measures the direction and strength of the linear

relationship between two quantitative variables.

Correlation is usually written as r.

1

1i i

x y

x x y xr

n s s


Classifying Fossils. Exercise 3.19 (Page 138) gives the

lengths of two bones in five fossil specimens of extinct

beast Archaeopteryx.

: 38 56 59 64 74

: 41 63 70 72 84

Femur

Humerus

A. Find the correlation r step-by-step. That is find the mean

and standard deviation of the femur lengths and of the

humerus lengths. Then find the five standardized values

for each variable and use the formula for r.


38 41 1.53 1.573 2.4079

56 63 0.1667 0.1888 0.03147

59 70 0.0601 0.25173 0.01526

64 72 0.43944 0.37759 0.16593

74 84 1.1971 1.1328 1.356

3.976594

i i i i

x y x y

x x y y x x y yx y

s s s s

58.2x 13.1985xs 66y 15.89025ys


1

1i i

x y

x x y xr

n s s

1

3.976 0.9944

r

Facts About Correlation (r)

Correlation makes no distinction between

explanatory and response variable.

Correlation requires both variables to be

quantitative.

Correlation (r) itself has no unit of measurement;

its just a number.

Positive (r) indicates positive association.

Negative (r) indicates negative association.

Facts About Correlation (r)

The correlation (r) is always between -1 and 1.

The closer (r) is to +1, the stronger the evidence of positive association between two variables.

The closer (r) is to -1, the stronger the evidence of negative association between to variables.

If (r) is close to 0, does not rule out any strong relationship between x and y, there could still be a strong relationship but one that is not linear.

Correlation is strongly effected by a outlying observations.

Positive Linear Correlation

Perfect Positive

Linear Correlation

Strong Positive

Linear Correlation

1r 0.9r

Weak Positive

Linear Correlation

0.4r

Negative Linear Correlation

Perfect Negative

Linear Correlation

Strong Negative

Linear Correlation

1r 0.9r

Weak Negative

Linear Correlation

0.4r

No Linear Correlation

No Linear Correlation

r is close to 0

Describing the Strength

of a Linear Relationship

0 10.80.5– 1 – 0.8 – 0.5

Str

ong

Str

ong

Modera

te

Mo

dera

te

Weak

Example – Page 149, #3.32Do people with larger brains have higher IQ scores? A

study looked at 40 volunteer subjects, 20 men and 20

women. Brain size was measured by magnetic resonance

imagining. Table 3-3 gives the data. the MRI count is the

number of “pixels” the brain covered in the image. IQ was

measured by the Wechsler test.

A) Make a scatterplot of IQ score versus MRI count, using

distinct symbols for the mean and women. In addition

find the correlation between IQ and MRI for all 40 subjects

for the men alone and for the women alone.


0 3576

0 4984

0 3257

.

.

.

all

men

women

r

r

r


B) Men are larger than women on the average, so they

have larger brains. How is this size effect visible in your

plot? Find the mean MRI count for men and women to verify

the difference.

954 855

862 655

,

,

men

women

x

x

The points for men are

generally located on the right

side of the plot , while the women’s

points are generally on the left.

Example – Page 149, #3.32C) Your result in (b) suggests separating men and women in

looking at he relationship between brain size and IQ.

Use your work in (a) to comment on the nature and

strength of this relationship for women and for men.


The correlation for men and women suggests that there is

a moderately positive association for men and a weak one

for women. However, one significant feature of the data

that can be observed in the scatterplot is that the sample

group was highly stratified; that is, there were 10 men and

10 women with high IQs (at least 130), while other 10 of

each gender had IQs of no more than 103. The men’s

higher correlation can be attributed partly to the two

subjects with large brains and 103 IQs (which are relative

to the low IQ group). The men’s correlation might not

remain so high with a larger sample.

Lesson 3-3

Line of Best Fit

Best Fit Line

Is a straight line (equation) that describes how a response variable y changes as an explanatory variable x changes.

A best Fit Line (equation) is used to predict the value of y for a given value of x.

Best Fit Line unlike correlation, requires that we have an explanatory variable and a response variable.

Best Fit Line Is the line that comes closer

to all the points.

y is the actual value

is the predicted value.

Residual is the difference between the observed value and the associated predicted value.

Negative residual shows a model that is overestimate

Positive residual shows a model value that is underestimate.

Residualy

“Best Fit” Means Least-Squares

The line of “Best Fit” is

the line for which the

sum of the squared

residuals is the smallest.

The line of “Best Fit” is

called a Least-squares

Regression Line

(LSRL)

Equation of LSRL

The line must go through the point

Equation

Slope

Intercept

0 1y b b x

1

y

x

sb r

s

0 1b y b x

,x y


Keeping water supplies clean requires regular measurement

of levels of pollutants. The measurements are indirect – a

typical analysis involves forming a dye by a chemical

reaction with the dissolved pollutant, then passing light

through the solution and measuring its “absorbance.”

To calibrate such measurements, the laboratory measures

known standard solutions and uses regression to relate

series of data on the absorbance for different levels of

nitrates. Nitrates are measured in milligrams per liter of water


50 50 100 200 400 800 1200 1600 2000 2000

7.0 7.5 12.8 24.0 47.0 93.0 138.0 183.0 230.0 226.0

Nitrates

Absorbance

A. Chemical theory says that these data should lie on a

straight line. If the correlation is not at least 0.997,

something went wrong and the calibration procedure is

repeated. Plot the data and find the correlation. Must the

calibration be done again.840

802.704

96.83

90.953

N

N

A

A

x

s

y

s

0.9999

Recalibration is not necessary

840 96.831

10 1 802.704 90.953i ix y

r


B. What is the equation of the least-square lines for

predicating absorbance from concentration? If the lab

analyzed a specimen with 500 milligrams of nitrates per

liter, what do you expect the absorbance to be? Based

on your plot and the correlation, do you expect your

predicted absorbance to be very accurate?

840

802.704

96.83

90.953

N

N

A

A

x

s

y

s

0.9999r

1

90.953(0.9999) 0.1133

802.704

y

x

sb r

s

0 1

0

0

96.83 (0.1133)(840)

1.658

b y b x

b

b

ˆ 1.658 0.1133y x


ˆ 1.658 0.1133y x

500

ˆ 1.658 0.1133(500)

58.31

x

y

This prediction should be

very accurate since the

relationship is so strong

Lesson 3-3

The Role of r2 in Regression

Assessing the Accuracy of

the Linear Model The standard deviation and r² are numerical measures use

in the assessment of how well the model fits.

r² - Coefficient of Determination Measure of the proportion of variability in the y variable that can be

“explained” by the linear relationship between x and y.

100 r² is the percentage of variation in y that can be attributed to approximate linear relationship between x and y.

Is a number between 0 and 1, the close r² is to 1, the better the line describes how the change in the explanatory variable affects the value of the response variable.

r² is the square of the linear correlation coefficient for the least-square regression model.

A natural measure of variation about the least-square regression line is the sum of the squared residuals. Measures the amount of variation in y that cannot be explained by

the linear relationship between x and y.

Coefficient of Determination

y

x

Total

Deviation

y y

,x y

ˆ,x y

Unexplained

Deviation y a bx ˆy y

Explained

Deviation

y y

y

Coefficient of Determination

r² =Explained Variation

Total Variation

= 1 –Total Variation

Unexplained Variation

22

2

2

ˆy y y ySST SSEr

SST y y

Total Unexplained

Total


A study of class attendance and grades among first

year students at a state university showed that in

general students who attended a higher percent

of their classes earned higher grades. Class attendance

explained 16% of the variation in grade index among

students. What is the numerical value of the correlation

between percent of classes attended and grade index?

2 0.16r

2 0.16

0.40

r

r

Higher attendance goes with

high grades, so the correlation

must be positive


Some people think that the behavior of the stock

market in January predicts its behavior for the rest of

the year. Take the explanatory variable x to be the

percent change in a stock market index January and the

response variable y to be the change in the index for

the entire year. We expect a positive correlation between

x and y because the change in January contributes to

the full year’s change. Calculation from data for the years

1960 to 1997 gives

1.75%

9.07%

x

y

5.36%xs

15.35%ys

0.596r


1.75%

9.07%

x

y

5.36%xs

15.35%ys

0.596r

A. What percent of the observed variation in yearly

changes in the index is explanatory by a straight-line

relationship with the change during January.

2 2

0.596

(0.596) 0.355 35.5%

r

r


1.75%

9.07%

x

y

5.36%xs

15.35%ys

0.596r

B. What is the equation of the least-squares line for

predicting full-year change from January change?

1

0.1535(0.596) 1.707

.0536

y

x

sb r

s

0 1

0

0

.0907 (1.707)(.0175)

0.06083 6.083%

b y b x

b

bˆ 6.083% 1.707y x

Example – Page 166, #3.441.75%

9.07%

x

y

C. The mean change in January is . Use your

regression line to predict the change in the index in a

year in which the index rises 1.75% in January. Why

could you have given this result (up to roundoff error)

without doing the calculation.

ˆ 6.083% 1.707y x

1.75%x

The predicted change is . Since the

regression line must pass through .

9.07%y

,x y

Assumptions and Conditions

Check the scatterplot. The shape must be linear or we can’t use regression

at all.

Watch out for outliers.Outlying values have large residuals and squaring

makes their influences that much greater.

Outlying points can dramatically change a regression model. They can change the sign of the slope, misleading us about

the underlying relationship between variables.

A r² of 100% You may have accidentally regressed two variables

that measure the same thing.

Lesson 3-3

Residual Plot

Assumptions and Conditions Don’t extrapolate beyond the data.

A linear model will often do reasonable job of

summarizing a relationship in the narrow range of

observed x-values.

Beware of predicting y-values for x-values that lie

outside the range of the original data.

If you must extrapolate into the future, at least don’t

believe that the prediction will come true!

Don’t infer that x causes y just because there is

good linear model for their relationship.

Residuals

A residuals is the difference between an

observed value of the response variable and the

value predicted by the regression line.

Unexplained Deviation

Plotting the residual

A residual plot is a scatterplot of the (x, residual)

pairs

Residual plot is a good place to start when

assessing the appropriateness of the regression

line.

Residual Plot

Determine whether a linear model is appropriate to describe the relationship between the explanatory and response variables.

Residual are what is “left over” after the model describes the relationship, they often reveal subtleties that were not clear from a plot of the original data.

Determine whether the variance of the residuals is constant.

Check for outliers

Residual Plots – Uniformed

The uniform scatter of points indicated that the regression

line is good model

Residual Plots – Curved

The residuals have a curved pattern, so a straight line

is an inappropriate model

Residual Plots – Increasing/Decreasing

The response variable y has more spread for larger values

of the explanatory variable x, so prediction will be less

accurate when x is large.

Residual Plot – Individual Points

Individual points with large

residual (Child 19) are outliers.

Individual points that are extreme

in the x direction (Child 18)

are influential observations

Influential Observations

An outlier is an observation that lies outside the overall

pattern of the other observation.

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 directions of a scatterplot

are often influential for the least-square regression line.

Case 1

(outlier)

Case 2

Case 3

(influential)

Influential observations typically exist when

the point is large relative to its x value.


Lean body mass as a predictor of metabolic rate. Exercise

3.12, page 132 provides data from a study of dieting for

12 women and 7 men subjects. We explore the data

further.

A. Define two lists on your calculator, (L1) MASSF for

female mass and (L2) METF for female metabolic

rate. Define plots using the plotting symbol, and

plot the scatterplot.


B. Perform least-squares regression on your calculator and

record the equation and the correlation. Lean body mass

explains what percent of the variation in metabolic rate

for women?

ˆ 201.1616 24.026y x

2

.8765

0.7682 76.8%

r

r

Lean body mass explains about 76.82% of the variation

in metabolic rate.


C. Does the least-square line provide an adequate model

for the data? Define Plot2 to be a residual plot on your

calculator with residuals on the vertical axis and the

lean body mass (x-values) on the horizontal axis. Use

the plotting symbol.


Graphing Residuals

Y1 → Vars/Y-Vars/Function


Turn on Stat Plot 1

ZoomStat and Graph


C. From the residual plot, the line does appear to provide

an adequate model. The residual are scattered about

the horizontal axis and no patterns are evident.

D. Define L3 to be the predicated y-values: define PLOT3

to be a residual plot on your calculator with residuals

on the vertical axis and predicted metabolic rate on

the horizontal axis. Use the + plotting symbol.


D.The plots are identical in terms of the relative positions

of the data points.

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