# Units of measurement and the t-statistic

Consider a case in which we want to examine the effect of house value on homeupkeep expenditure using linear regression model, where dependent variable y is the upkeepexpense and independent variable x is home value. Suppose that units of y and x are both in US dollar. Let $$b_{1}$$ be the slope estimate and $$t_{1}$$ be the t-statistic value in testing $$H_{0}: β_{1}= 0$$.Now we convert both y and x into Canadian dollar. Let $$b_{1}^{*}$$ be the new slope estimate and $$t_{1}^{*}$$ be the t statistic value. Show that $$t_{1}^{*}$$ = $$t_{1}$$

• Please add the self-study tag and read its wiki – kjetil b halvorsen Feb 20 at 18:25
• Observe that $ax/ay = x/y$, and think about how that might apply to the formula for calculating the t-statistic. – jbowman Feb 20 at 18:27
• I'm more confused about the effect of the change in units on the standard error of $\beta_{1}$ – Nicholas Heinrich Feb 20 at 18:30
• The standard error is in the same units as the underlying values. – whuber Feb 20 at 18:32
• So if $a$ was the scalar representing the change to canadian dollars, it would be $t_{1}^{*}$ = $\frac{b_{1}^{*}}{se(b_{1}^{*})}$ = $\frac{ab_{1}}{ase(b_{1})}$ = $\frac{b_{1}}{se(b_{1})}$ = $t_{1}$ ??? – Nicholas Heinrich Feb 20 at 18:38