Linearising using Logarithms (DP IB Applications & Interpretation (AI): HL): Revision Note

Be careful with the different constants here!

For this note $a$ and $b$ are the constants as found in the exponential model $y=a{b}^x$.

When you find the regression line in Step 2, your GDC will probably refer to it in the form $y=ax+b$, but those are not the same $a$ and $b$!

When you use your GDC to find the regression line of Y on X, $Y=mX+c$:

• the gradient $m$ will be the GDC's $a$

• the $y$-intercept $c$ will be the GDC's $b$

Hatter has noticed that over the past 50 years there seem to be fewer hatmakers in London. He also knows that global temperatures have been rising over the same time period. He decides to see if there could be any correlation, so he collects data on the number of hatmakers and the global mean temperatures from the past 50 years and records the information in the graph below.

Hatter suggests that the equation for $h$ in terms of $t$ can be written in the form $h=a{b}^t$
. He linearises the data using $x=t$  and $y=\ln h$ and calculates the regression line of $y$ on $x$ to be $y=4.382-1.005x$.

 Find the values of $a$ and $b$.

Answer:

Be careful with the different constants here!

For this section $a$ and $b$ are the constants as found in the power model $y=a{x}^b$.

When you find the regression line in Step 2, your GDC will probably refer to it in the form $y=ax+b$, but those are not the same $a$ and $b$!

When you use your GDC to find the regression line of Y on X, $Y=mX+c$:

• the gradient $m$ will be the GDC's $a$

• the $y$-intercept $c$ will be the GDC's $b$

The graph below shows the heights, $h$ metres, and the amount of time spent sleeping, $t$ hours, of a group of young giraffes. It is believed the data can be modelled using $t=a{h}^b$
.

The data are coded using the changes of variables $x=\ln h$ and $y=\ln t$. The regression line of $y$ on $x$ is found to be $y=0.3-1.2x$.

Find the values of $a$ and $b$.

Answer: