Download The data for a Y-on-X regression problem come in the form (x1, Y1

ASSUMPTIONS IN A REGRESSION MODEL
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The data for a Y-on-X regression problem come in the form (x1, Y1), (x2, Y2), …., (xn, Yn).
These may be conveniently laid out in a matrix or spreadsheet:
Case
1
2
.
.
n
x
x1
x2
.
.
xn
Y
Y1
Y2
.
.
Yn
The word “case” might be replaced by “point” or “data point” or “sequence number” or
might even be completely absent. The labels x and Y could be other names, such as
“year” or “sales.” In a data file in Minitab, the values for the x’s and Y’s will be actual
numbers, rather than algebra symbols. In an Excel spreadsheet, these could be either
numbers or implicit values.
If a computer program is asked for the regression of Y on x, then numeric calculations
will be done. These calculations have something to say about the regression model,
which we discuss now.
The most common linear regression model is this.
The values x1 , x 2 ,..., x n are known non-random quantities which are measured
without error. If in fact the x values really are random, then we assume that they
are fixed once we have observed them. This is a verbal sleight of hand;
technically we say we are doing the analysis “conditional on the x’s.”
The Y-values are independent of each other, and they are related to the x’s through
the model equation
Yi = 0 + 1 xi + i for i = 1, 2, 3, …, n
The symbols 0 and 1 in the model equation are nonrandom unknown
parameters.
The symbols 1,2, …, n are called “statistical noise” or “errors.” The -values
prevent us from seeing the exact linear relationship between x and Y. These
-values are unobserved random quantities. They are assumed to be statistically
independent of each other, and they are assumed to have expected value zero. It
is also assumed that (using SD for standard deviation) SD(1) = SD(2) = … =
SE(n) = . The symbol  is another nonrandom unknown parameter.
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ASSUMPTIONS IN A REGRESSION MODEL
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The calculations that we will do for a regression will make statements about the model.
S
For example, the estimated regression slope b1 = xy is an estimate of the parameter 1.
Sxx
Here is a summary of a few regression calculations, along with the statements that they
make about the model.
Calculation
b1 =
What it means
Sxy
Estimate of regression slope 1
Sxx
b0 = y - b1 x
Estimate of regression intercept 0
Residual mean square
Estimate of 2
Root mean square residual (standard error
of regression, standard error of estimate)
Estimate of 
Standard error of an estimated coefficient
t (of an estimated coefficient)
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Estimate of the standard deviation of that
coefficient
Estimated coefficient, divided by its
standard error
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