The accompanying data resulted from an experiment in which weld diameter \(x\)
and shear strength \(y\) (in pounds) were determined for five different spot
welds on steel. A scatterplot shows a pronounced linear pattern. With
\(\sum(x-\bar{x})=1000\) and \(\sum(x-\bar{x})(y-\bar{y})=8577\),
the least-squares line is \(\hat{y}=-936.22+8.577 x\).
\(\begin{array}{llllll}x & 200.1 & 210.1 & 220.1 & 230.1 & 240.0\end{array}\)
\(\begin{array}{llllll}y & 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2\end{array}\)
a. Because \(1 \mathrm{lb}=0.4536 \mathrm{~kg}\), strength observations can be
re-expressed in kilograms through multiplication by this conversion factor:
new \(y=0.4536(\) old \(y) .\) What is the equation of the least-squares line when
\(y\) is expressed in kilograms?
b. More generally, suppose that each \(y\) value in a data set consisting of
\(n(x, y)\) pairs is multiplied by a conversion factor \(c\) (which changes the
units of measurement for \(y\) ). What effect does this have on the slope \(b\)
(i.e., how does the new value of \(b\) compare to the value before conversion),
on the intercept \(a\), and on the equation of the least-squares line? Verify
your conjectures by using the given formulas for \(b\) and \(a\). (Hint: Replace
\(y\) with \(c y\), and see what happens
\- and remember, this conversion will affect \(\bar{y} .\) )
