Question

one month, he examined his text message and call logs with his closest friends. for each friend, mike checked the number of text messages he sent to that friend, x, and the number of minutes they spoke on the phone, y. round your answer to the nearest thousandth. r =

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(x\) be the number of text - messages and \(y\) be the number of minutes on the phone.
\(\bar{x}=\frac{13 + 40+52+89+96}{5}=\frac{290}{5} = 58\)
\(\bar{y}=\frac{94+70+90+75+58}{5}=\frac{387}{5}=77.4\)

Step2: Calculate the numerator and denominator of the correlation coefficient formula

The formula for the correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)

• Calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\):
• For \(x = 13,y = 94\): \((13 - 58)(94 - 77.4)=(-45)\times16.6=-747\)
• For \(x = 40,y = 70\): \((40 - 58)(70 - 77.4)=(-18)\times(-7.4) = 133.2\)
• For \(x = 52,y = 90\): \((52 - 58)(90 - 77.4)=(-6)\times12.6=-75.6\)
• For \(x = 89,y = 75\): \((89 - 58)(75 - 77.4)=31\times(-2.4)=-74.4\)
• For \(x = 96,y = 58\): \((96 - 58)(58 - 77.4)=38\times(-19.4)=-737.2\)
• \(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=-747+133.2-75.6 - 74.4-737.2=-1501\)
• Calculate \((x_{i}-\bar{x})^{2}\):
• For \(x = 13\): \((13 - 58)^{2}=(-45)^{2}=2025\)
• For \(x = 40\): \((40 - 58)^{2}=(-18)^{2}=324\)
• For \(x = 52\): \((52 - 58)^{2}=(-6)^{2}=36\)
• For \(x = 89\): \((89 - 58)^{2}=31^{2}=961\)
• For \(x = 96\): \((96 - 58)^{2}=38^{2}=1444\)
• \(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=2025 + 324+36+961+1444=4790\)
• Calculate \((y_{i}-\bar{y})^{2}\):
• For \(y = 94\): \((94 - 77.4)^{2}=16.6^{2}=275.56\)
• For \(y = 70\): \((70 - 77.4)^{2}=(-7.4)^{2}=54.76\)
• For \(y = 90\): \((90 - 77.4)^{2}=12.6^{2}=158.76\)
• For \(y = 75\): \((75 - 77.4)^{2}=(-2.4)^{2}=5.76\)
• For \(y = 58\): \((58 - 77.4)^{2}=(-19.4)^{2}=376.36\)
• \(\sum_{i = 1}^{5}(y_{i}-\bar{y})^{2}=275.56+54.76+158.76+5.76+376.36=871.2\)

Step3: Calculate the correlation coefficient \(r\)

\(r=\frac{-1501}{\sqrt{4790\times871.2}}\)
First, \(4790\times871.2 = 4790\times871.2=4172048\)
\(\sqrt{4172048}\approx2042.56\)
\(r=\frac{-1501}{2042.56}\approx - 0.735\)

Answer:

\(-0.735\)