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Chapter 4: Getting in Line with Simple Linear Regression
By writing y = 3.69 + 0.113x, you mean that this equation represents your
estimated value of y, given the value of x that you observe with your data.
Statisticians write this equation by using a carrot (or hat as statisticians call
/
it), like y, so everyone can know it’s an estimate, not the actual value of y.
This y-hat is your estimate of the average value of y over the long term, based
on the observed values of x. However, in many intro stats texts, the hat is left
off because statisticians have an unwritten understanding as to what y repre-
sents. This issue comes up again in Chapters 5 through 8. (By the way, if you
think y-hat is a funny term here, it’s even funnier in Mexico, where statisti-
cians call it y-sombrero — no kidding!)
The y-intercept of the regression line
Selected parts of that Minitab output shown in Figure 4-2 are of importance to
you at this point. First, you can see that under the column “Coef” you have
the numerical values on the right side of the equation of the line — in other
words, the slope and y-intercept. The number 3.69 represents the coefficient 75
of “Constant,” which is a fancy way of saying that’s the y-intercept (because
the y-intercept is just a constant, it never changes). The y-intercept is the
point where the line crosses the y-axis, in other words, the value of y when
x equals 0.
The y-intercept of a regression line may or may not have a practical meaning
depending on the situation. To determine whether the y-intercept of a regres-
sion line has practical meaning, look at the following:
Does the y-intercept fall within the actual values in the data set? If yes,
then it has practical meaning.
Does the y-intercept fall into negative territory where negative y-values
aren’t possible? For example if the y-values, are weights, they can’t be
negative. Then the y-intercept has no practical meaning. It is still correct
though, because it just happens to be the place where the line, if
extended to the y-axis, crosses the y-axis.
Does the value x = 0 have practical meaning? For example, if x is temper-
ature at a football game in Green Bay, then x = 0 is a value that’s relevant
to examine. If x = 0 has practical meaning, then the y-intercept would
also because it represents the value of y when x = 0. If not, for example,
when x represents height of a toddler, then the y-intercept has no practi-
cal meaning.
In the textbook example, the y-intercept doesn’t really have a practical mean-
ing because students don’t weigh zero pounds, so you don’t really care what
the estimated textbook weight is for that situation. But you do need to find a
line that fits the data you do have (where average student weights go from