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Chapter 4: Getting in Line with Simple Linear Regression
If the relationship between x and y is uphill, or positive (as x increases so
does y), the correlation is a positive number. If the relationship is downhill, or
negative (as x increases, y gets smaller), then the correlation is negative. If
the correlation is zero, you can find no linear relationship between x and y.
(It may be that a different relationship exists, such as a curve; see Chapter 7
for more on this.)
If the value of the correlation is +1 or –1, this value indicates that the points
fall in a perfect, straight line. If the correlation is close to +1 or –1, this corre-
lation value signifies a strong relationship. If the correlation is closer to +0.5
or –0.5, these values show a moderate relationship. A value close to 0 signi-
fies a weak relationship or no linear relationship at all.
You can calculate the correlation coefficient by using a formula involving the
standard deviation of x, the standard deviation of y, and the covariance of x
and y, which measures how x and y move together, in relation to their means.
However, the formula isn’t the focus here (you can find it in your intro stats
text or in my other book Statistics For Dummies [Wiley]); it’s the concept 73
that’s important. Any computer package can calculate the correlation coeffi-
cient for you with a simple click of the mouse.
To have Minitab calculate a correlation for you, go to Stat>Basic Statistics>
Correlation. Highlight the variables you want correlations for and click Select.
Then click on OK.
The correlation for the textbook weight example is (can you guess before
looking at it?) 0.926, which is very close to 1.0. This correlation means that a
very strong linear relationship is present between average textbook weight
and average student weight for grades 1 through 12, and that relationship is
positive and linear (follows a straight line). This correlation is confirmed by
the scatterplot shown in Figure 4-1.
Building a Simple Linear
Regression Model
After you have a handle on which x variables may be related to y in a linear
way, you go about the business of finding that straight line that best fits the
data. You find the slope and y-intercept, put them together to make a line,
and you use the equation of that line to make predictions for y. All of this is
part of building a simple linear regression model.
In this section, you set the foundation for regression models in general
(including those you can find in Chapters 5 through 8). You plot the data,
come up with a model that you think makes sense, assess how well it fits,
and use it to guesstimate the value of y given another variable, x.