Question

1. the table shows the heights and the lengths of several rectangles. what does the correlation coefficient for the data indicate about the strength of the linear association between the height and the length of these rectangles?

Explanation:

Step1: Calculate the correlation coefficient

We can use the formula for the correlation coefficient \(r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\)

Let \(x\) be the height and \(y\) be the length.

First, calculate \(\sum x = 41+70+21+34+10+92+54+24+10+35+42+66=509\)

\(\sum y=21 + 25+32+16+45+40+23+45+35+21+14+66 = 383\)

\(\sum xy=(41\times21)+(70\times25)+(21\times32)+(34\times16)+(10\times45)+(92\times40)+(54\times23)+(24\times45)+(10\times35)+(35\times21)+(42\times14)+(66\times66)\)

\(=861+1750+672+544+450+3680+1242+1080+350+735+588+4356\)

\(=16308\)

\(\sum x^{2}=41^{2}+70^{2}+21^{2}+34^{2}+10^{2}+92^{2}+54^{2}+24^{2}+10^{2}+35^{2}+42^{2}+66^{2}\)

\(=1681+4900+441+1156+100+8464+2916+576+100+1225+1764+4356\)

\(=27779\)

\(\sum y^{2}=21^{2}+25^{2}+32^{2}+16^{2}+45^{2}+40^{2}+23^{2}+45^{2}+35^{2}+21^{2}+14^{2}+66^{2}\)

\(=441+625+1024+256+2025+1600+529+2025+1225+441+196+4356\)

\(=14783\)

\(n = 12\)

\(r=\frac{12\times16308-509\times383}{\sqrt{[12\times27779 - 509^{2}][12\times14783-383^{2}]}}\)

\(=\frac{195696-195947}{\sqrt{[333348-259081][177396 - 146689]}}\)

\(=\frac{-251}{\sqrt{[74267][30707]}}\)

\(=\frac{-251}{\sqrt{2279633269}}\)

\(r\approx - 0.0016\)

Since \(|r|\approx0.0016\) which is very close to \(0\), the correlation is weak. And since \(r\approx - 0.0016<0\), it is a weak negative correlation

Answer:

Weak negative correlation