Question

consider the four data sets shown in the accompanying table. complete parts (a) and (b).
click the icon to view the data table.
(a) compute the linear correlation coefficient for each data set.
the linear correlation coefficient for the first data set is ( r = - 0.817 ).
(round to three decimal places as needed.)
the linear correlation coefficient for the second data set is ( r=square )
(round to three decimal places as needed.)

Explanation:

Step1: Calculate necessary sums for Data Set 2

Let \(n = 11\) (number of data points).
First, calculate \(\sum x\), \(\sum y\), \(\sum x^{2}\), \(\sum y^{2}\), \(\sum xy\) for Data Set 2.
\(\sum x=8 + 10+5+9+7+4+12+14+6+11+13 = 99\)
\(\sum y=9.04 + 7.95+8.58+9.81+9.33+10.96+8.24+5.26+11.84+5.82+6.68 = 89.51\)
\(\sum x^{2}=8^{2}+10^{2}+5^{2}+9^{2}+7^{2}+4^{2}+12^{2}+14^{2}+6^{2}+11^{2}+13^{2}=8^{2}+10^{2}+5^{2}+9^{2}+7^{2}+4^{2}+12^{2}+14^{2}+6^{2}+11^{2}+13^{2}=967\)
\(\sum y^{2}=9.04^{2}+7.95^{2}+8.58^{2}+9.81^{2}+9.33^{2}+10.96^{2}+8.24^{2}+5.26^{2}+11.84^{2}+5.82^{2}+6.68^{2}=797.9391\)
\(\sum xy=(8\times9.04)+(10\times7.95)+(5\times8.58)+(9\times9.81)+(7\times9.33)+(4\times10.96)+(12\times8.24)+(14\times5.26)+(6\times11.84)+(11\times5.82)+(13\times6.68)=797.09\)

Step2: Use the formula for the correlation coefficient \(r\)

The formula for the correlation coefficient is \(r=\frac{n\sum xy-\sum x\sum y}{\sqrt{n\sum x^{2}-(\sum x)^{2}}\sqrt{n\sum y^{2}-(\sum y)^{2}}}\)
Substitute \(n = 11\), \(\sum x = 99\), \(\sum y=89.51\), \(\sum xy = 797.09\), \(\sum x^{2}=967\), \(\sum y^{2}=797.9391\) into the formula.
First, calculate the numerator:
\(n\sum xy-\sum x\sum y=11\times797.09-99\times89.51=11\times797.09 - 99\times89.51=8767.99-8861.49=-93.5\)
Then, calculate \(n\sum x^{2}-(\sum x)^{2}=11\times967 - 99^{2}=10637-9801 = 836\)
\(n\sum y^{2}-(\sum y)^{2}=11\times797.9391-(89.51)^{2}=8777.3301 - 8011.0401=766.29\)
\(\sqrt{n\sum x^{2}-(\sum x)^{2}}\sqrt{n\sum y^{2}-(\sum y)^{2}}=\sqrt{836}\times\sqrt{766.29}\approx28.91\times27.68 = 800.23\)
\(r=\frac{-93.5}{800.23}\approx - 0.117\)

Answer:

\(-0.117\)