Chapter 3Chapter 3

Examining RelationshipsExamining Relationships

IntroductionIntroduction

• We have looked at only one-variable We have looked at only one-variable statistics: Quantitative & Categorical statistics: Quantitative & Categorical datadata

• Now, we examine how one variable Now, we examine how one variable affects another.affects another.

• These variables are called These variables are called RESPONSIVE and EXPLANATORY.RESPONSIVE and EXPLANATORY.

Two VariablesTwo Variables

• Response Variables – measure the Response Variables – measure the outcome of a study (dependent outcome of a study (dependent variables).variables).

• Explanatory Variables – explains the Explanatory Variables – explains the observed outcome (independent observed outcome (independent variables).variables).

• Since responsive variables depend on Since responsive variables depend on explanatory variables, how variables are explanatory variables, how variables are labeled depend on how the data is used. labeled depend on how the data is used.

Page 122: Exercise 3.1Page 122: Exercise 3.1

• EXPLANATORYEXPLANATORY • RESPONSIVERESPONSIVE

a) Study Time a) Study Time Grade Grade

b) Explore the relationship … neither b) Explore the relationship … neither explanatory. explanatory.

c) Yearly Rainfallc) Yearly Rainfall Crop Yield Crop Yield

d) Explore the relationship … neither d) Explore the relationship … neither explanatory.explanatory.

e) Occup. class of dade) Occup. class of dad Occup. Occup. class of son class of son

Page 123: Exercise 3.2Page 123: Exercise 3.2

• Explanatory – Height @ Age 6Explanatory – Height @ Age 6

• Responsive – Height @ Age 16Responsive – Height @ Age 16

• Both QuantitativeBoth Quantitative

Guide to Examine DataGuide to Examine Data

• Plot and add numerical summariesPlot and add numerical summaries

• Look for overall patterns and any Look for overall patterns and any deviationsdeviations

• When the overall pattern is regular, When the overall pattern is regular, use a mathematical model to use a mathematical model to describe what is occurring.describe what is occurring.

3.1 SCATTERPLOTS3.1 SCATTERPLOTS

ActivityActivity

Scatterplots ReviewScatterplots Review

• A scatterplot shows the relationship A scatterplot shows the relationship between two quantitative variables between two quantitative variables measured on the same individuals.measured on the same individuals.

• Once we plot the variables, we look Once we plot the variables, we look for a pattern called correlation. for a pattern called correlation.

• If a correlation exists, we use a If a correlation exists, we use a mathematical model to explain the mathematical model to explain the pattern.pattern.

ScatterplotsScatterplots

The PlotThe Plot• Place explanatory variables, x, (if one Place explanatory variables, x, (if one

exists) on the horizontal axis and exists) on the horizontal axis and responsive, y, on the vertical.responsive, y, on the vertical.

• If there isn’t an explanatory-response If there isn’t an explanatory-response relationship, either variable may go relationship, either variable may go on the horizontal axis.on the horizontal axis.

• Read the tips on Page 128.Read the tips on Page 128.

LionsLions, , Tigers,Tigers, && Bears … Oh My! Bears … Oh My!

Let’s look at a study of the weight of a bear vs. Let’s look at a study of the weight of a bear vs. the chest size of a bear.the chest size of a bear.

A biologist feels that the size of a bear’s chest A biologist feels that the size of a bear’s chest can predict the weight of a bear. Which is the can predict the weight of a bear. Which is the response variable? The explanatory variable?response variable? The explanatory variable?

Chest Chest size size (in)(in)

2626 4545 5454 4949 4141 4949 4444 1919

WeighWeight (lb)t (lb)

9090 344344 416416 348348 262262 360360 332332 3434

Bear DataBear Data

• Make a scatterplot.Make a scatterplot.

• Save your lists.Save your lists.

EquationEquation

• Diagnostics OnDiagnostics On

• Weight = 11.27 chest - 187.46Weight = 11.27 chest - 187.46• Meaning: a = 11.27 slope – increase of 11.27 lbs in Meaning: a = 11.27 slope – increase of 11.27 lbs in

weight for every additional inch of chest size. b = weight for every additional inch of chest size. b = -187.46 is the y-intercept – for a bear with a chest size -187.46 is the y-intercept – for a bear with a chest size of 0 inches, the weight would be -187.46 lbs. Reality: of 0 inches, the weight would be -187.46 lbs. Reality: this number has no significance. It is a numerical value this number has no significance. It is a numerical value to help estimate the regression line. to help estimate the regression line.

Bear Data AssociationBear Data Association

• Does it appear a bear’s weight is Does it appear a bear’s weight is explained by its chest size?explained by its chest size?

• What is the shape of the dots?What is the shape of the dots?

• Is it a positive/negative association?Is it a positive/negative association?

• How strong is the relationship?How strong is the relationship?

• Are there any outliers?Are there any outliers?

Practice: Exercise 3.6Practice: Exercise 3.6

(a)(a) Number of powerboats and Number of powerboats and number of manatees killed by number of manatees killed by boats. Which is the explanatory boats. Which is the explanatory variable?variable?

(b)(b) Make a scatterplot. What does the Make a scatterplot. What does the scatterplot show about this scatterplot show about this relationship?relationship?

PLOTPLOTa) Number of Powerboat Registrations.b) Plot

Powerboat Registrations (1000)

ManateesKilled

Interpreting ScatterplotsInterpreting Scatterplots

• Graph – look for the overall pattern Graph – look for the overall pattern and any striking deviations, outliers.and any striking deviations, outliers.

• Descriptions: Form, direction, and Descriptions: Form, direction, and strength.strength.– Strength is determined by how close the Strength is determined by how close the

points follow a clear form.points follow a clear form.

Association – Positive or Association – Positive or NegativeNegative

• Positive association – above-average Positive association – above-average values of one variable accompany values of one variable accompany above-average values of the other or above-average values of the other or below-average values of one yield below-average values of one yield below-average values of the other.below-average values of the other.

• Negative association – one variable is Negative association – one variable is above-average and the other below above-average and the other below or vice-versa.or vice-versa.

Categorical VariablesCategorical Variables

• When adding on categorical variables When adding on categorical variables to a scatterplot, use different to a scatterplot, use different symbols, colors, to plot points.symbols, colors, to plot points.

• Homework: Exercises 3.3, 3.4, 3.7, Homework: Exercises 3.3, 3.4, 3.7, 3.9 – 3.12 (Exercise 3.13 allows you 3.9 – 3.12 (Exercise 3.13 allows you to expedite things with 3.11 with to expedite things with 3.11 with added Calc fun.)added Calc fun.)