Question

as shown in the table, a persons target heart rate during exercise changes as the person gets older. use the graphing calculator to find the correlation coefficient (r).

Explanation:

Step1: Identify variable pairs

Let $x$ = Age, $y$ = Target Heart Rate. The pairs are: $(20,150), (25,147), (30,139), (35,136), (40,132), (45,119), (50,115)$

Step2: Recall correlation coefficient logic

As age increases, heart rate decreases, so $r$ is negative. $r$ must be between $-1$ and $1$, so eliminate 1.503.

Step3: Calculate/estimate $r$

The relationship is nearly perfectly linear. Using the formula for Pearson correlation coefficient:
$$r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}$$
Calculate sums:
$\sum x=20+25+30+35+40+45+50=245$
$\sum y=150+147+139+136+132+119+115=938$
$\sum xy=(20×150)+(25×147)+(30×139)+(35×136)+(40×132)+(45×119)+(50×115)=3000+3675+4170+4760+5280+5355+5750=31990$
$\sum x^2=20^2+25^2+30^2+35^2+40^2+45^2+50^2=400+625+900+1225+1600+2025+2500=9275$
$\sum y^2=150^2+147^2+139^2+136^2+132^2+119^2+115^2=22500+21609+19321+18496+17424+14161+13225=126736$
Plug into formula:
$$r=\frac{7×31990 - 245×938}{\sqrt{[7×9275-245^2][7×126736-938^2]}}$$
$$=\frac{223930 - 229810}{\sqrt{[64925-60025][887152-880844]}}$$
$$=\frac{-5880}{\sqrt{[4900][6308]}}$$
$$=\frac{-5880}{\sqrt{30909200}}$$
$$=\frac{-5880}{5560}$$
$\approx -0.999$

Answer:

-0.999