This PowerPoint helps students to consider the concept of infinity.
An Overview of Simple Linear Regression
1. Regression Analysis
Explain the impact of one variable on another
Independent variable:
the variable used to explain
the dependent variable
Dependent
variable
the variable you
wish to explain
is used to
x-axis
y-axis
Simple Linear
independent
Annual Salary vs. Education
$0
$20,000
$40,000
$60,000
$80,000
$100,000
$120,000
0 2 4 6 8 10
Annual salary
Years of Education (past high school)
?
2. Regression Analysis
Independent variable
Dependent variable
is also used to
x-axis
y-axis
Simple Linear
Quantify linear relationships
…develop an equation for the…
Y = mX + b
value of Y when X = 0
change in Y relative to a change in X
Y = b1X + b0
slope
intercept
The purpose of regression analysis is calculate estimates of the slope and intercept.
3. Regression Analysis
Independent variable
Dependent variable
is also used to
x-axis
y-axis
Simple Linear
Quantify linear relationships
…develop an equation for the…
Y = mX + b
value of Y when X = 0
change in Y relative to a change in X
Y = b1X + b0
slope
intercept
The purpose of regression analysis is calculate estimates of the slope and intercept.
=INTERCEPT(y-range, x-range)
=SLOPE(y-range, x-range)
using LEAST SQUARES ESTIMATION
4. Linear Regression Example
Scatterplot
House price model: scatter plot
0
50
100
150
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450
0 500 1000 1500 2000 2500 3000
Square Feet
House Price ($1000s)
(1) Describe
the
relationship…
r = 0.762
5. House price model: scatter plot and regression line
y = 0.1098x + 98.248
0
50
100
150
200
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350
400
450
0
500
1000
1500
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2500
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Price in $1000s
sqft
Selling Price vs. Square Feet
trendline
(2) Model the Data…
Calculate the regression coefficients
and interpret
b0 is the estimated mean value of Y when the value of X is 0
(if X = 0 is in the range of observed X values)
Because the square footage of the house cannot be 0, the Y intercept has no practical application.
slope
intercept
6. House price model: scatter plot and regression line
y = 0.1098x + 98.248
0
50
100
150
200
250
300
350
400
450
0
500
1000
1500
2000
2500
3000
Price in $1000s
sqft
Selling Price vs. Square Feet
trendline
(2) Model the Data…
Calculate the regression coefficients
and interpret
b1 measures the mean change in the average value of Y as a result of a one-unit change in X The mean value of a house increases by 0.1098($1000) = $109.80, on average, for each additional one square foot of size
slope
intercept
7. y = 0.1098x + 98.248 R² = 0.5808
0
50
100
150
200
250
300
350
400
450
0
500
1000
1500
2000
2500
3000
Price in $1000s
sqft
Selling Price vs. Square Feet
(3) Evaluate the Model…
Calculate R2
58% of the variability in the PRICE of a home is exaplined by using the SIZE of the home.
8. Standard Error of Estimate
The standard deviation of the
observations around the
regression line.
When predicting, this is the stdev
of the predictions
(3) Evaluate
the Model…
Calculate R2 and std error
9. y = 0.1098x + 98.248 R² = 0.5808
0
50
100
150
200
250
300
350
400
450
0
500
1000
1500
2000
2500
3000
Price in $1000s
sqft
Selling Price vs. Square Feet
(3) Evaluate the Model…
Calculate R2 and std error
SYX=41.33