Warm-up

An investigator wants to study the effectiveness of two surgical procedures to correct near-sightedness: Procedure A uses cuts from a scalpel and procedure B uses a laser. The data to be collected are the degrees of improvement in vision after the procedure is performed.

Design an experiment for this.

Comparitive Graphs

Section 1.1Creating and Interpreting Comparative Graphs

After this section, you should be able to…

CONSTRUCT and INTERPRET Comparative bar graphs

CONSTRUCT and INTERPRET Segmented bar graphs

CONSTRUCT and INTERPRET Two Way Tables

CALCULATE & INTERPRET marginal and conditional distributions

ORGANIZE statistical problems

Learning Objectives

Two Way Table

Describes two categorical variables.

One variable is shown in the rows and the other is in the columns.

Example of Two Way Table

Young adults by gender & chance of getting rich

  Gender  

Opinion Female Male Total

Almost no chance 96 98 194

Some chance but probably not 426 286 712

A 50-50 chance 696 720 1416

A godd chance 663 758 1421

Almost certain 486 597 1083

Total 2367 2459 4826

Reading a Two-Way Table

Look at the distribution of each variable separately.The totals on the right are strictly the values

for the distribution of opinions about becoming rich for all.

The totals at the bottom are for gender

Marginal Distribution

The marginal distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table.

It’s the distribution of each category alone.

Percentages

Often are more informative

Used when comparing groups of different sizes.

Find the percent of young adults who they there is a good chance they will be rich.

Young adults by gender & chance of getting rich

  Gender  

Opinion Female Male Total

Almost no chance 96 98 194

Some chance but probably not 426 286 712

A 50-50 chance 696 720 1416

A godd chance 663 758 1421

Almost certain 486 597 1083

Total 2367 2459 4826

Find the marginal distribution (in %) of opinions. Make a graph to display the marginal distribution.

Young adults by gender & chance of getting rich

  Gender  

Opinion Female Male Total

Almost no chance 96 98 194

Some chance but probably not 426 286 712

A 50-50 chance 696 720 1416

A godd chance 663 758 1421

Almost certain 486 597 1083

Total 2367 2459 4826

Response Percent

Almost no chance 4.0%

Some chance but probably not 14.8%

A 50-50 chance 29.3%

A good chance 29.4%

Almost certain 22.4%

Find the marginal distribution (in %) of gender. Make a graph to display the marginal distribution.

Young adults by gender & chance of getting rich

  Gender  

Opinion Female Male Total

Almost no chance 96 98 194

Some chance but probably not 426 286 712

A 50-50 chance 696 720 1416

A godd chance 663 758 1421

Almost certain 486 597 1083

Total 2367 2459 4826

Response Percent

Male 51%

Female 49%

Conditional Distribution

It describes the values of that variable among individuals who have a specific value of another variable.

To describe the relationship between the two categorical variables

Conditional Distribution of young women and men and their opinion.

Young adults by gender & chance of getting rich

  Gender

Opinion Female Male

Almost no chance 96 98

Some chance but probably not 426 286

A 50-50 chance 696 720

A godd chance 663 758

Almost certain 486 597

Total 2367 2459

Side-by-Side Bar Graph

Response Women Men

Almost no chance 4.1% 4%

Some chance but probably not 18.0% 11.6%

A 50-50 chance 29.4% 29.3%

A good chance 28% 30.8%

Almost certain 20.5% 24.3%

Segmented Bar Graph

Response Women Men

Almost no chance 4.1% 4%

Some chance but probably not 18.0% 11.6%

A 50-50 chance 29.4% 29.3%

A good chance 28% 30.8%

Almost certain 20.5% 24.3%

Did we look at the right conditional distribution? Our goal was to analyze the relationship

between gender and opinion about chances of becoming rich for these young adults.

Hint: Does gender influence opinion or opinion influence gender?

Since gender influences opinion, then we want to consider the conditional distribution of opinion for each gender.

Four-Step Process State: What’s the question that you’re

trying to answer? Plan: How will you go about answering

the question? What statistical techniques does this problem call for?

Do: Make graphs and carry out needed calculations.

Conclude: Give your practical conclusion in the setting of the real-world problem.

State

What is the relationship between gender and responses to the question “What do you think are the chances you will have much more than a middle-class income at age 30?”

Plan

We suspect that gender might influence a young adult’s opinion about the chance of getting rich. So we’ll compare the conditional distributions of response for men alone and for women alone.

Response Women Men

Almost no chance 4.1% 4%

Some chance but probably not 18.0% 11.6%

A 50-50 chance 29.4% 29.3%

A good chance 28% 30.8%

Almost certain 20.5% 24.3%

Do

We’ll make a side-by side bar graph to compare the opinions of males and females.

I could have used a segmented as well!

Side-by Side Comparative Bar Graph

Response Women Men

Almost no chance 4.1% 4%

Some chance but probably not 18.0% 11.6%

A 50-50 chance 29.4% 29.3%

A good chance 28% 30.8%

Almost certain 20.5% 24.3%

Segmented Comparative Bar Graph

Response Women Men

Almost no chance 4.1% 4%

Some chance but probably not 18.0% 11.6%

A 50-50 chance 29.4% 29.3%

A good chance 28% 30.8%

Almost certain 20.5% 24.3%

Conclude Based on the sample data, men seem

somewhat more optimistic about their future income than women. Men were less likely to say that they have “some chance but probably no” than women (11.6% vs 18.0%). Men were more likely to say that they have a “good chance” (30.8% vs 28.0%) aor alre “almost certain” (24.3% vs 20.5%) to have much more than a middle-class income by age 30 than women were.

Association

We say there is an association between two variables if specific values of one variable tend to occur in common with specific values of the other.Be careful though….even a strong association

between two categorical variables can be influenced by other variables lurking in the background.

Simpson’s Paradox

An association between two variables that holds for each individual value of a thrid variable can be changed or even reversed when the data for all values of the third variable are combined. This reversal is called Simpson’s paradox.

Accident victims are sometimes taken by helicopter from the accident scene to a hospital. Helicopters save taim. Do they also save lives?

  Helicopter Road

Victim Died 64 260

Victim survived 136 840

Total 200 1100

32% of helicopter patients died, but only 24% of the others did. This seems discouraging!

Helicopter is sent mostly to serious accidents.

Serious Accident

  Helicopter Road

Died 48 60

Survived 52 40

Total 100 100

Less Serious Accident

  Helicopter Road

Died 16 200

Survived 84 800

Total 100 1000

Titanic Disaster

Homework

Page 24 (19, 21, 23, 24, 25, 27-32, 33, 35, 36)