Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 4 Regression Analysis:...

Copyright © 2014 Pearson Education, Inc. All rights reserved

Chapter 4

Regression Analysis:

Exploring Associations

between Variables

4 - 2 Copyright © 2014 Pearson Education, Inc. All rights reserved

Learning Objectives

Be able to write a concise and accurate description of an association between two continuous variables based on a scatterplot.

Understand how to use a regression line to summarize a linear association between two continuous variables.

Interpret the intercept and slope of a regression line in context and know how to use the regression line to predict mean values of the response variable.

Critically evaluate a regression model.


4.1

Visualizing Variability with a Scatterplot


Scatterplots

Used to investigate a positive, negative, or no association between two numerical variables.

In states where women tend to marry at an older age, men also tend to marry at an older age.


Positive Trend

Older cars tend to have more miles than newer cars.

Newer cars tend to have fewer miles than older cars.

There is a positive association between car age and miles the car has been driven.


Negative Trend

Countries with higher literacy rates tend to have fewer births per woman.

Countries with lower literacy rates tend to have more births per woman.

There is a negative association between literacy rate and births per woman.


No Trend

There is no trend between the speed and age of a marathon runner.

Knowing the age of a marathon runner does not help predict the runner’s speed.

There is no association between a marathon runner’s age and speed.


Strength of Association

If for each value of x, there is a small spread of y values, then there is a strong association between x and y.

If for each value of x, there is a large spread of y values, then there is a weak or no association between x and y.

If there is a strong (weak) association between x and y, then x is a good (bad) predictor of y.


Strength of Association


Linear Trends

A trend is linear if there is a line such that the points in general do not stray far from the line.

Linear trends are the easiest to work with. There is a positive linear association between number

of searches for “Vampire” and number for “Zombie”.


Other Shapes

Nonlinear association can also occur, but this is covered in a more advanced statistics course.

Only use techniques from this chapter when there is a linear trend.


Summary of Analysis of the Scatterplot

Look to see if there is a trend or association. Determine the strength of trend. Is the

association strong or weak? Look at the shape of the trend. Is it linear?

Is it nonlinear?


Writing Clear Descriptions Based on Association

Good: People who have higher salaries tend to travel farther

on vacation. A person who has a high salary is predicted to travel

far on vacation.

Bad: Because they have higher salaries, they travel farther. A person with a high salary will travel farther on

vacation.


4.2

Measuring Strength of Association with

Correlation


The Correlation Coefficient r

The correlation coefficient is a number, r, that measures the strength of the linear association between two variables.

-1 ≤ r ≤ 1 If r is close to 1, then there is a strong positive linear

association. If r is close to -1, then there is a strong negative

linear association. If r is close to 0, then there is a weak or no

association.


Positive Correlation


Weak or No Correlation


Negative Correlation


Interpreting Correlation

The correlation between daily swim suits and ski jackets purchased in an apparel store is r = -0.96

There is a strong negative correlation between daily swim suits and ski jackets purchased.

On days with strong swim suit sales, one predicts that ski jacket sales would be weak.

This does not mean that people who buy swim suits are causing potential ski jacket buyers to not buy.


Using StatCrunch to Find r

Enter Data Stat→Regression→

SimpleLinear Select the Variables Calculate


Switching x and y

r for life expectancy Women vs. Men: r = 0.977 r for life expectancy Men vs. Women: r = 0.977 Switching x and y has no effect on r.


Correlation, Arithmetic, and Units

Multiplying all x’s or all y’s by a constant does not change r.

Adding the same constant to all x’s or all y’s does not change r.

Changing units such as in→cm or ºF→ºC does not change r.

r is unitless.


Correlation and Linearity and Outliers

Only use linear correlation to interpret the data when there is a linear relationship

An outlier can strongly influence the correlation.


4.3

Modeling Linear Trends


Least Squares Regression Line

The Regression Line is the “best fit” line for the data.

The line minimizes the average squared vertical distances.

It is only useful with data with a linear model.


StatCrunch and the Regression Line

Enter the data. Select x and y variables. Stat→Regression

→Simple Linear Select: Plot the Fitted

Line Calculate


Interpreting the Slope

The slope is the coefficient in front of x in the regression line equation.

Rise/Run means that if x is increased by 1, then y is predicted or increases by an average of the slope value.

The slope is only meaningful if the data follows a linear model.


Interpreting the Slope

The slope is 1.2. If x is increased by 1, y has an average

increase of 1.2. For every $100 the company spends on ads,

it averages an additional $120 in revenue.


Interpreting the y-intercept

The y-intercept is the value of y when x is 0. Use the y-intercept to interpret the data only

when: It makes sense to have a value of 0 for x. The calculated y-intercept value is meaningful. The data include values equal to or close to 0.


Interpreting the y-intercept

The y-intercept is 1.8. If the company spends no money, $0, on

advertising, it is predicted to have an average daily revenue of $180.


Why Not to Use the y-intercept

A sample of high school freshmen and sophomores resulted in a regression equation that relates age to height in inches: predicted height = -9.2 + 4.9x

The y-intercept is -9.2. A height of -9.2 inches is meaningless. The sample only included teenagers. The age of 0 years

is too far from the ages in the sample. The slope is meaningful. High school freshmen and

sophomores grow an average of 4.9 inches per year.


Correlation is Not Causation

A strong correlation is not evidence of a cause-and-effect relationship.

Do not use the words, “causes”, “makes”, “will”, “because”, etc. when making regression analysis based conclusions.

Do use the words, “predict”, “tends”, and“on average”.


More on the Regression Line

The equation does change when x and y are switched.

If the linear model is a “good fit” for the data, then the mean value of y for a given x will nearly lie on the regression line.


4.4

Evaluating the Linear Model


Nonlinear Data

If you can’t imagine a line don’t try to find one.

If the association is not linear, don’t attempt to find or interpret r or the equation of the least squares regression line.


Slope and Causation

Predicted Salary = 22,000 + 8,000 College Years Wrong: Each year in college results in an

additional salary increase of $8,000. Wrong: A person with one more year of

college education will earn an extra $8,000. Correct: On average, people with one more

year of college education tend to earn an extra $8,000.


Beware of Outliers

Outliers have a strong effect on both the correlation and the equation of the regression line.

An outlier that strongly effects the regression line is called an influential point.

When there is an influential point present, perform regression analysis both with and without the influential point.


Example of an Influential Point


Regression of Aggregate Data

Using Aggregate Data for regression means that each point represents the mean of all the y-values with a given x-value.

When using aggregate data, be sure to include the word “mean” in all interpretations.


Aggregate Data

There is a weak correlation between math SAT scores and critical reading SAT scores.

There is a strong correlation between states’ mean math SAT scores and states’ mean critical reading SAT scores.


Don’t Extrapolate

Only use the regression line to predict y-values for x-values that are within or near the range of the data.

Predicted Height = 31.78 + 2.45 Age Predict for a 50 year old:

31.78 + 2.45(50) = 154.28 inches The predicted height of a 50 year old man is over

12 feet tall????


Coefficient of Determination r2

r2 measures how much of the variation in the response variable, y, can be explained by the explanatory variable, x.

r2 is used to help determine which explanatory variable would be best for making predictions about the response variable.


Coefficient of Determination Example

60.5% of the variation in the value of cars can be explained by the age of the car. The other 39.5% cannot be explained by the age of the car.


Chapter 4

Case Study


Scatterplot of City Government Income vs. Private Meter Income Without Brinks

Positive weak linear association Predicted Collection = 688497 + 145.5 (City Income)


Are Brinks Employees Stealing from Parking Meters?

New York City contracted Brinks to collect parking meter money. The city suspects that employees are keeping some of it.

There is data on the monthly meter collection of honest (not Brinks) collectors vs. the city’s total income for that month.


Predicted vs. Actual Brinks Collection

Predicted Collection = 688497 + 145.5 (City Income) One month, City Income was $7016 and Brinks

collected $1,330,143. 688497 + 145.5(7016) = $1,709,325 Discrepancy: 1,709,352 – 1,330,143 = $379,182


Comparing Brinks vs. Honest Employees

Conclusion: Income when Brinks is working is clearly lower than when the honest employees are working.


Chapter 4

Guided Exercise 1


Does the Cost of a Flight Depend on the Distance? How much would it

cost to fly 500 miles? Use a complete

regression analysis.


Create a Scatterplot

Since the cost tends to increase as mileage increases and since there is no apparent strong curvature, the linear model is appropriate.


The Regression Line

Interpret the Slope: 0.08. For every additional mile, on average, the price

goes up by $0.08. Interpret the y-intercept: 163

This is the predicted price for a 0 mile flight. The y-intercept is meaningless here.


Answer the Question

How much would it cost to fly 500 miles? Predicted Cost = 162.60 + 0.0796 (miles) 162.60 + 0.0796 (500) = 202.40 A 500 mile flight is predicted to cost $202.40.


Chapter 4

Guided Exercise 2


Test Scores: Slope

The summary statistics between the midterm and final exam scores are: Midterm: Mean = 75, Standard Dev. = 10 Final: Mean = 75, Standard Dev. = 10 r = 0.7, n = 20

First find the slope:

final

midterm

sb r

s

10

0.7 0.710


Test Scores: y-intercept

Midterm: Mean = 75, Standard Dev. = 10

Final: Mean = 75, Standard Dev. = 10

r = 0.7, n = 20, b = 0.7 Then find the y-intercept a from the equation:

a y bx

75 0.7(75) 22.5


Test Scores: Regression Line

Midterm: Mean = 75, Standard Dev. = 10

Final: Mean = 75, Standard Dev. = 10

r = 0.7, n = 20, b = 0.7, a = 22.5 Write out the following equation:

Predicted = a + bx Predicted Final Score = 22.5 + 0.7(Midterm Score) Use the equation to predict the final score for a midterm score of 95%.

Predicted Final = 22.5 + 0.7(95) = 89 This is less than 95 since the slope is less than 1.

Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 4 Regression Analysis:...

Documents

Transcript of Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 4 Regression Analysis:...