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### STAT E100

Section Week 3 - Regression

- Descriptive Statistics versus Hypothesis Testing
- Outliers
- Sample vs. Population
- Residual Plots

Residual = observed value (y) - predicted value (ŷ)

- Correlation ≠ Causation
- R2 is the coefficient of determination that tells the proportion of variability in y that can be explained by x.
- Leverage refers to data points whose x-values stand away from the mean of x. These points are said to exert leverage on a linear model. High-leverage points pull the line close to them, and so they can have a large effect on the line.

Sample Question #1

1) Data comes from: http://www.cbssports.com/mlb/salaries/teams.

Data was collected on the opening day payroll (in millions of $) and the number of wins that year for the 30 major league baseball (MLB) teams in 2010. Below are the histograms of the 2 variables:

a) Based on these histograms, describe the distribution of each of these variables. The scatterplot of y = wins vs. x = payroll is on the next slide.

Sample Question #1

1) Data comes from: http://www.cbssports.com/mlb/salaries/teams.

Data was collected on the opening day payroll (in millions of $) and the number of wins that year for the 30 major league baseball (MLB) teams in 2010. Below are the histograms of the 2 variables:

Based on these histograms, describe the distribution of each of these variables. The scatterplot of y = wins vs. x = payroll is on the next slide.

It seems that the distributions are more or less symmetrical (although there is a slight skewness in the variable wins). Also, note the isolated point on the payroll histogram.

Sample Question #1

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

b) What do you think the correlation coefficient is between these 2 variables?

Here is some more SPSS output:

Sample Question #1

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

b) What do you think the correlation coefficient is between these 2 variables?

Here is some more SPSS output:

Correlation coefficient = 0.536

Sample Question #1

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

c) What is the equation for the best fit line for this data?

For your reference (probably should show how to run the regression and correlation in SPSS):

d) What is the estimated number of wins for a team with a payroll of $120.1 million (the Red Sox)? How would this be predicted to change if it increased by $10 million?

Sample Question #1

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

c) What is the equation for the best fit line for this data?

For your reference (probably should show how to run the regression and correlation in SPSS):

d) What is the estimated number of wins for a team with a payroll of $120.1 million (the Red Sox)? How would this be predicted to change if it increased by $10 million?

Sample Question #1

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

c) What is the equation for the best fit line for this data?

ywins = 0.167 xpayroll + 67.982

For your reference (probably should show how to run the regression and correlation in SPSS):

d) What is the estimated number of wins for a team with a payroll of $120.1 million (the Red Sox)? How would this be predicted to change if it increased by $10 million?

Sample Question #1 d) What is the estimated number of wins for a team with a payroll of $120.1 million (the Red Sox)? How would this be predicted to change if it increased by $10 million?

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

c) What is the equation for the best fit line for this data?

ywins = 0.167 xpayroll + 67.982

For your reference (probably should show how to run the regression and correlation in SPSS):

ywins = 0.167 (120.1) + 67.982 = 88.0387 88 wins

If increased by 10 million, then the numbers of wins would increase becase the slope is positive.

Sample Question #1

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

e) The Red Sox actually won 86 games in 2010. Calculate their residual.

f) What is R2 for this model? What is its interpretation?

g) Based on the scatterplot, do you have any concerns for the using this regression model?

h) How the regression line change if the Yankees were removed (payroll = $194.66 million, wins = 97)?

Sample Question #1

Data comes from: http://www.cbssports.com/mlb/salaries/teams.

e) The Red Sox actually won 86 games in 2010. Calculate their residual.

Residual = 88.0387 2 wins

f) What is R2 for this model? What is its interpretation?

R2 = (0.536) 2 = 0.287 The regression line explains 28.7% of the variability in wins.

g) Based on the scatterplot, do you have any concerns for using this regression model?

There may be a point with high residual.

h) How would the regression line change if the Yankees were removed (payroll = $194.66 million, wins = 97)?

This point may have high residual.

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