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STAT 3220 Midterm 1 Introduction To
Regression Analysis 2025/
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simple linear regression model
assumes that the relationship between the dependent variable and the independent variable can be approximated by a straight line
y = β₀+β₁x+ε
it is reasonable to describe the relationship between y and x by using the simple linear regression model if...
the y values tend to inc or dec in a straight-line fashion as the x values inc
dependent variable
response, y
independent variable
predictor, x
μ(y|x)=β₀+β₁
- mu of y given x
- a y-intercept of Beta zero and a slope of Beta one
- aka "line of means"
- the mean value of y when the value of independent variable is x
ε
error term, epsilon
describes the effect on y of all factors other than the actual x variable
time series data
when data are observed in time sequence
cross-sectional data
data observed at a single point in time
least squares prediction equation
y-hat = b₀+b₁x
mean value
the average of all the values of the dependent variable that could potentially be observed when the independent variable equals a particular value
simple coefficient of determination
r²
a measure of the usefulness of a simple linear regression model
total variation
- the sum of squared prediction errors obtained when we do not employ the predictor variable x
- measures total amt of variation exhibited by the observed values of y
unexplained variation
SSE
- the sum of squared prediction errors obtained when we use the predictor variable x
- measures amt of variation in the values of y that is not explained bu the predictor variable
explained variation
total variation - unexplained variation
r²
= explained variation/total variation
- the variation explained by the simple linear regression model
simple correlation coefficient
r
point estimate of rho
measures the strength of the linear relationship between y and x
population correlation coefficient
ρ (rho)
all possible combinations of observed values of x and y
H₀: ρ=0 vs Ha: ρ≠
there is no linear relationship between x and y vs there is a linear relationship between x and y
F-test
H₀: β₁=0 vs Ha: β₁≠
testing the significance of the simple linear regression model
