## 35.11 Summary

In this chapter,
we have learnt about **regression**,
which mathematically describes the relationship between two *quantitative* variables.
The response variable is denoted by \(y\),
and the explanatory variable by \(x\).
The linear relationship between them
(the **regression equation**),
in the sample, is

\[
\hat{y} = b_0 + b_1 x,
\]
where \(b_0\) is a number (the **intercept**),
\(b_1\) is a number (the **slope**),
and the ‘hat’ above the \(y\) indicates that the equation gives an
*predicted* mean value of \(y\) for the given \(x\) value.

The *intercept* is the predicted mean value of \(y\) when the value of \(x\) is zero.
The *slope* is how much the predicted mean value of \(y\) changes,
on average,
when the value of \(x\) *increases* by 1.

The regression equation can be used to make *predictions*
or to *understand* the relationship between the two variables.
Predictions made with values of \(x\) outside the values of \(x\) used to create the
regression equation
(called *extrapolation*)
may not be reliable.

In the population, the regression equation is

\[
\hat{y} = \beta_0 + \beta_1 x.
\]
To test a hypothesis about a population slope \(\beta_1\),
based on the value of the sample slope \(b_1\),
**assume** the value of \(\beta_1\) in the null hypothesis (usually zero) to be true.
Then,
the sample slope varies from sample to sample and,
under certain statistical validity conditions,
varies with an approximate normal distribution
centered around the hypothesised value of \(\beta_1\),
with a standard deviation of
\(\text{s.e.}(b_1)\).
This distribution describes what values of the sample slope
could be **expected** in the sample
if the value of \(\beta_1\) in the null hypothesis was true.
The *test statistic* is

\[
t = \frac{ b_1 - \beta_1}{\text{s.e.}(b_1)},
\]
where \(\beta_1\) is the hypothesised value given in the null hypothesis (usually zero).
The \(t\)-value is like a \(z\)-score,
and so an approximate **\(P\)-value** can be estimated using the
68–95–99.7 rule.

The following short video may help explain some of these concepts: