Exam 2- Regression Flashcards
Y=a+bx
Y-hat reminds us that we have deviations about the line and that values for y specified by the line are PREDICTIOnS a - intercept b - slope ^ Y- predicted value if y for a given x
Statistical model
An equation that fits the pattern between a response variable and possible explanatory variables, accounting for deviations from the model. Or in other words, a regression line
What does y intercept tells us?
The value of y when x=0
What does slope tell us?
The change in y for every one unit increase in x , on average!
As x increases by one unit what happened to the y when slope is negative?
Y decreases
As x increases by one unit what happens to y when slope is positive?
Y increases by rise/run units
b=
Rise(y)/run(x)
Interpretation of slope : rise/run
For every inch increase in height at age 4 , height increases by 1.15 inches ON AVERAGE at age 18
Interpretation of y- intercept
Males who are zero inches tall at age 4 will be 23 inches tall at age 18
The intercept is the value of y when x=O
How to predict
- collect data
- plot data
- predict
- fit the data with a straight line equation
- evaluate the equation
Residuals
Vertical distance from the observed y value and the line , or
The difference between observed y value and y-hat , the value predicted by regression line
Squared Prediction error (residual)2
(Observed y - predicted y)2= (Y - Y(hat)) squared
They are squared because the sum of two residuals are normally equals to zero ( negative residual plus positive residuals above and below the line)
Positive residuals
Points above the line
Negative residuals
Points below the line
The least-squares residual line is
The line with the smallest sum of squares errors (denoted SSE)
Sum of Squared Deviations (residuals, errors (SSE) represents
The total variation in observed values of y
Sum residuals2( squared) =
( y - y-hat) squaredLeast - squares equation
Y-hat=a +bx
Formula for a (intercept)
a=y-bar - bx(bar)
Where y and x are the respective means
Formula for b(slope)
Slope is a rate of change, the amount of change in y for a given value of x when x increases by 1
b=r Sy/Sx
Least-squares regressions line facts
- makes the distance of the data points from the line small Only in Y direction
- if we reverse the roles of two variables we get different least squared regression line
What is the connection between correlation r and the slope b of the least squared line?
Slope and r have the same sign B=r only when Sy=Sx Both r and b tell us the direction If r=0 b =O If ro b>0 If we know sign of r we know sign of b and vise versa
What b and r have in common
Always have the same sign
A change of 1 standard deviation in x corresponds to a change of r standard deviations in y.
Change in y(hat) is less then change in x
The least squares regression line always passes
Through the point (x bar;y bar)
Correlation r describes
The straight line relationship
