Copyright ©2011 Brooks/Cole, Cengage Learning Relationships Between Quantitative Variables Chapter 3 1.

Copyright ©2011 Brooks/Cole, Cengage Learning

Relationships Between

Quantitative Variables

Chapter 3

1

Copyright ©2011 Brooks/Cole, Cengage Learning 2

Principle Idea:

The description and confirmation of relationships between variables are very important in research.


Three Tools we will use …

• Scatterplot, a two-dimensional graph of data values

• Correlation, a statistic that measures the strength and direction of a linear relationship between two quantitative variables.

• Regression equation, an equation that describes the average relationship between a quantitative response and explanatory variable.


3.1 Looking for Patterns with Scatterplots

Questions to Ask about a Scatterplot

• What is the average pattern? Does it look like a straight line, or is it curved?

• What is the direction of the pattern?

• How much do individual points vary from the average pattern?

• Are there any unusual data points?


Positive/Negative Association/Linear Relationship• Two variables have a positive association when

the values of one variable tend to increase as the values of the other variable increase.

• Two variables have a negative association when the values of one variable tend to decrease as the values of the other variable increase.

• Two variables have a linear relationship when the pattern of their relationship resembles a straight line.


Example 3.1 Height and Handspan

Data shown are the first 12 observations of a data set that includes the heights (in inches) and fully stretched handspans (in centimeters) of 167 college students.


Example 3.1 Height and Handspan

Taller people tend to have greater handspan measurements than shorter people do.

When two variables tend to increase together, we say that they have a positive association.

The handspan and height measurements may have a linear relationship.


Example 3.2 Driver Age and MaximumLegibility Distance of Highway Signs

• A research firm determined the maximum distance at which each of 30 drivers could read a newly designed sign.

• The 30 participants in the study ranged in age from 18 to 82 years old.

• We want to examine the relationship between age and the sign legibility distance.



• We see a negative association with a linear pattern.

• We will use a straight-line equation to model this relationship.


Example 3.3 The Development of Musical Preferences

• The 108 participants in the study ranged in age from 16 to 86 years old.

• We want to examine the relationship between song-specific age (age in the year the song was popular) and musical preference (positive score above average, negative score below average).


Example 3.3 The Development of Musical Preferences

• Popular music preferences acquired in late adolescence and early adulthood.

• The association is nonlinear.


Groups and Outliers

• Use different plotting symbols or colors to represent different subgroups.

• Look for outliers: points that have an usual combination of data values.


3.2 Describing Linear Patterns with a Regression Line

Two purposes of the regression line:• to estimate the average value of y at any

specified value of x• to predict the value of y for an individual,

given that individual’s x value

When the best equation for describing the relationship between x and y is a straight line, the equation is called the regression line.


Example 3.5 Height and Handspan (cont)

Based on line, at height of 60 inches tall, handspan is about 18 cm,at height of 70 inches tall, handspan is about 21.5 cm.So change in height of 10 inches corresponds to change in handspan of about 3.5 cm 3.5/10 = 0.35 cm per inch.Estimated slope is about 0.35 cm per inch.

Scatterplot with ‘best’ regression line (via Minitab)


The Equation for the Regression Line

is spoken as “y-hat,” and it is also referred to either as predicted y or estimated y.

b0 is the intercept of the straight line. The intercept is the value of y when x = 0.

b1 is the slope of the straight line. The slope tells us how much of an increase (or decrease) there is for the y variable when the x variable increases by one unit. The sign of the slope tells us whether y increases or decreases when x increases.

xbby 10ˆ y


Example 3.6 Writing the Regression Equation for Height and Handspan

Regression equation: Handspan = -3 + 0.35 Height

Estimate the average handspan for people 60 inches tall:Average handspan = -3 + 0.35(60) = 18 cm.

Predict the handspan for someone who is 60 inches tall:Predicted handspan = -3 + 0.35(60) = 18 cm.

Note: in a statistical relationship, there is variation from the

average pattern.


Interpreting the y-Intercept and the Slope

b0 = -3 is the y-intercept, the estimated or predicted handspan for someone whose height (x) is 0 inches. No meaningful interpretation in this example.

b1 = 0.35 is the slope, we estimate the handspan increases by 0.35 cm, on average, for each increase of 1 inch in height.

Regression equation: Handspan = -3 + 0.35 Height


Regression equation: Distance = 577 - 3 Age


Estimate the average distance for 20-year-old drivers:Average distance = 577 – 3(20) = 517 ft.

Predict the legibility distance for a 20-year-old driver:Predicted distance = 577 – 3(20) = 517 ft.

Slope of –3 tells us that, on average, the legibility distance decreases 3 feet when age increases by 1 year.


Prediction Errors and Residuals

• Prediction Error = difference between the observed value of y and the predicted value .

• Residual =

y

yy ˆ


Regression equation: = 577 – 3x

Example 3.8 Prediction Errors for the Highway Sign Data

Can compute the residual for all 30 observations.Positive residual observed value higher than predicted.Negative residual observed value lower than predicted.

x = Age y = Distance Residual

18 510 577 – 3(18)=523 510 – 523 = -13

20 590 577 – 3(20)=517 590 – 517 = 73

22 516 577 – 3(22)=511 516 – 511 = 5

xy 3577ˆ

y


The Least Squares Estimation Criterion

Least Squares Regression Line: minimizes the sum of squared prediction errors.

SSE: Sum of squared prediction errors.

Formulas for Slope and Intercept:

ii

iii

xx

yyxxb

21

xbyb 10


3.3 Measuring Strength and Direction with Correlation

• The strength of the relationship is determined by the closeness of the points to a straight line.

• The direction is determined by whether one variable generally increases or generally decreases when the other variable increases.

Correlation r indicates the strength and the direction of a straight-line relationship.


Interpretation the Correlation Coefficient

• r is always between –1 and +1• magnitude indicates the strength• r = –1 or +1 indicates a perfect linear relationship• sign indicates the direction• r = 0 indicates a slope of 0 so knowing x does not

change the predicted value of y

• Formula for correlation:

y

i

x

i

s

yy

s

xx

nr

1

1


Example 3.10 Correlation Between Handspan and Height

Regression equation: Handspan = -3 + 0.35(Height)

Correlation r = +0.74

a somewhat strong positive linear relationship.


Regression equation: Distance = 577 – 3(Age)

Example 3.11 Correlation Between Age and Sign Legibility Distance

Correlation r = -0.8 a somewhat strong negative linear association.


Example 3.12 Left and Right Handspans

If you know the span of a person’s right hand, can you accurately predict his/her left handspan?Correlation r = +0.95 a very strong positive linear relationship.


Example 3.13 Verbal SAT and GPAGrade point averages (GPAs) and verbal SAT scores for a sample of 100 university students.

Correlation r = 0.485

a moderately strong positive linear relationship.


Example 3.14 Age and Hours of TVWatching per Day

Relationship between age and hours of daily television viewing for 1299 survey respondents.

Correlation r = 0.136 a weak connection.Note: a few claimed to watch more than 24 hours/day!


Example 3.15 Hours of Sleep and Hours of Study

Relationship between reported hours of sleep the previous 24 hours and the reported hours of study during the same period for a sample of 116 college students.

Correlation r = –0.36 a not too strong negative association.


Interpretation of and Formula for r2

Squared correlation r2 is between 0 and 1 and indicates the proportion of variation in the response explained by x.

SSTO = sum of squares total = sum of squared differences between observed y values and .

SSE = sum of squared errors (residuals) = sum of squared differences between observed y values and predicted values based on least squares line.

SSTO

SSESSTOr

2

y


Interpretation of r2

Height and Right Handspanr2 = 0.55 Height explains 55% of the variation among observed right handspans

TV viewing and Ager2 = 0.0185 only about 1.85%; knowing a person’s age doesn’t help much in predicting amount of daily TV viewing.


Reading Computer Results for Regression


3.4 Regression and Correlation Difficulties and Disasters

• Extrapolating too far beyond the observed range of x values

• Allowing outliers to overly influence results

• Combining groups inappropriately

• Using correlation and a straight-line equation to describe curvilinear data


Extrapolation

• Risky to use a regression equation to predict values far outside the range where the original data fell (called extrapolation).

• No guarantee that the relationship will continue beyond the range for which we have observed data.


Example 3.17 Height and Foot Length

Regression equation uncorrected data: 15.4 + 0.13 heightcorrected data: -3.2 + 0.42 height

Correlationuncorrected data: r = 0.28corrected data: r = 0.69

Three outliers were data entry errors.


Example 3.18 Earthquakes in US

Correlationall data: r = 0.26w/o SF: r = –0.824


Example 3.19 Height and Lead Feet

Scatterplot of all data: College student heights and responses to the question “What is the fastest you have ever driven a car?”

Scatterplot by gender:Combining two groups led to illegitimate correlation


Example 3.20 U.S. Population Predictions

Correlation: r = +0.96Regression Line: population = –2348 + 1.289(Year)Poor Prediction for Year 2030 = –2348 + 1.289(2030) or about 269 million, due to curved (not linear) pattern.

Population of US (in millions) for each census year between 1790 and 2000.


3.5 Correlation Does Not Prove Causation

1. Causation2. Confounding Factors Present3. Explanatory and Response are both

affected by other variables4. Response variable is causing a change

in the explanatory variable

Interpretations of an Observed Association


Case Study 3.1 A Weighty Issue

Relationship between Actual and Ideal Weight

Females Males

Copyright ©2011 Brooks/Cole, Cengage Learning Relationships Between Quantitative Variables Chapter 3 1.

Documents

cengage learning relationships

average relationship

linear pattern

average pattern

driver age

variable increase

songspecific age age

positive association