Statistics Solutions

[Step-by-Step] Sometimes it is known in advance that a line must go through the origin, that is, that the equation of the line is y = bx . In this case,

Question: Sometimes it is known in advance that a line must go through the origin, that is, that the equation of the line is

y = bx .

In this case, finding the least squares line reduces to finding the value b that minimizes the equation

\[f(b)=\sum\limits_{i=1}^{n}{{{\left( {{y}_{i}}-b{{x}_{i}} \right)}^{2}}.}\]
  1. Show that the derivative of f with respect to b is given by
    \[{{f}^{'}}(b)=-2\sum\limits_{i=1}^{n}{{{x}_{i}}{{y}_{i}}+2b\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}.}}\]
  2. Use the expression for \[{{f}^{'}}(b)\] computed in part (a) to show that the least squares line has slope
\[\hat{b}=\frac{\sum\limits_{i=1}^{n}{{{x}_{i}}{{y}_{i}}}}{\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}}\]

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