Chapter 2 Examining Relationships

Response variable measures outcome of a study (dependent variable)

Explanatory variable explains or influences changes in response variable (independent)

Practice: pg 81 #2.1, 2.2

2.1 Scatterplots

Scatterplot

shows relationship between two quantitative variables measured on the same individuals

most common way to display relation between two variables

Example using calculator: pg 83 #2.4 powerboats vs. manatees killed

Examining a scatterplot

Look for overall pattern and deviations from the pattern

Describe overall shape by the form, direction, and strength of the relationship

Look for outliers

Clusters

Look for: positively and negatively associated strength is determined by how closely

the points follow a clear form (use strong or weak)

Adding categorical variables to scatterplots use different colors or symbols to plot points

Practice: pg. 89 #2.7

Section 2.1 practice problems: pg 91 #2.8, 9, 10, 11

2.2 Correlation

Correlation

A linear relationship is strong if the points lie close to a straight line. Our eyes are not good judges of strength of linear relationship

measures the direction and strength of the linear relationship between two quantitative variables

equation: to calculate on calculator: calculate with

manatees

))((11

yx syy

sxx

nr

Correlation r is the average product of z-scores r is positive when there is a positive

association r requires that both variables be

quantitative r has no unit of measure its just a

number

r is strongly affected by outliers r is used along with the mean and

standard deviation to describe two-variable data

Section 2.2 practice problems pg 103 2.21, 24, 29

2.3 Least-Squares Regression

least-squares regression line

regression line is a straight line that describes how a response variable (y) changes as an explanatory variable (x) (best fit line)

standard line that doesn’t depend on our guess as to where the line should go

line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible

to calculate the equation of regression line with calculator:

Facts about least-squares regression

the least-squares regression looks at distance from y to line (not x) so you cannot switch the x and y variables and get the same line

the least-squares regression line always passes through the point (x,y)

the square of the correlation is the proportion of data accounted for by the least-square regression line when you report a regression, give r2 as a measure

of how successful the regression was in explaining the response

Practice problems: pg 114 #2.33, 34

Residuals the difference between an observed value of

the response variable and the value predicted by the regression line (y-ŷ)

to calculate residuals on calculator: examining residuals helps assess how well the

line describes the data the mean of the residuals is always zero residual plot is a scatterplot of the regression

residuals against the explanatory variable to help assess the fit of a regression line

Make residual plot of manatees data

Things to look for in residual plot

1. Curved pattern – shows relationship is not linear

2. Increasing or decreasing spread – prediction of y when x is large will be less accurate

3. Individual points with large residuals – these points are outliers

4. Individual points that are extreme in the x direction

Consider the following data on x=height (in inches) and y=average weight (in pounds) for American females, age 30-39

X 58 59 60 61 62 63 64 65 66 67 Y 113 115 118 121 124 128 131 134 137 141

X 68 69 70 71 72 Y 145 150 153 159 164

Make scatterplot. Is the data linear? Plot residual plot.

Practice: pg 122 #2.36, 37Section 2.3 practice: pg 124 # 2.42, 44, 45, 47, 48

2.4 Cautions about Correlation and Regression

Extrapolation

use of a regression line for prediction far outside the range of values of the explanatory variable (x) that you used to obtain the line

often is not accurate

Using averaged data

average temp, average, salary, average age, etc.

do not apply results to individuals correlations based on averages are

usually too high when applied to individuals

Lurking variables

sometimes the relationship between two variables is influenced by other variables that were not measured or even thought about

Association is not causation

a strong association does not necessarily mean that x causes y

example: manatee deaths vs powerboat registrations

Section 2.4 practice: pg 136 2.57, 58, 59