Question

rounded to three decimal places, what is the value of r for this data set?
variable x 8 15 3 7 2 14 20
variable y 24 41 10 22 8 40 59
0.972
0.995
0.998
0.951

Explanation:

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

The mean of \(x\) values: \(\bar{x}=\frac{8 + 15+3+7+2+14+20}{7}=\frac{69}{7}\approx9.857\)
The mean of \(y\) values: \(\bar{y}=\frac{24 + 41+10+22+8+40+59}{7}=\frac{204}{7}\approx29.143\)

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 each \(i\):

• For \(x = 8,y = 24\): \((8 - 9.857)(24-29.143)=(- 1.857)(-5.143)\approx9.55\)
• For \(x = 15,y = 41\): \((15 - 9.857)(41 - 29.143)=(5.143)(11.857)\approx60.99\)
• For \(x = 3,y = 10\): \((3 - 9.857)(10 - 29.143)=(-6.857)(-19.143)\approx131.33\)
• For \(x = 7,y = 22\): \((7 - 9.857)(22 - 29.143)=(-2.857)(-7.143)\approx20.43\)
• For \(x = 2,y = 8\): \((2 - 9.857)(8 - 29.143)=(-7.857)(-21.143)\approx166.11\)
• For \(x = 14,y = 40\): \((14 - 9.857)(40 - 29.143)=(4.143)(10.857)\approx45.07\)
• For \(x = 20,y = 59\): \((20 - 9.857)(59 - 29.143)=(10.143)(29.857)\approx302.86\)

Sum of \((x_{i}-\bar{x})(y_{i}-\bar{y})\): \(9.55+60.99 + 131.33+20.43+166.11+45.07+302.86=736.34\)

Calculate \((x_{i}-\bar{x})^{2}\) for each \(i\):

• \((8 - 9.857)^{2}\approx3.45\)
• \((15 - 9.857)^{2}\approx26.45\)
• \((3 - 9.857)^{2}\approx47.02\)
• \((7 - 9.857)^{2}\approx8.16\)
• \((2 - 9.857)^{2}\approx61.73\)
• \((14 - 9.857)^{2}\approx17.16\)
• \((20 - 9.857)^{2}\approx102.88\)

Sum of \((x_{i}-\bar{x})^{2}\): \(3.45+26.45+47.02+8.16+61.73+17.16+102.88 = 266.85\)

Calculate \((y_{i}-\bar{y})^{2}\) for each \(i\):

• \((24 - 29.143)^{2}\approx26.45\)
• \((41 - 29.143)^{2}\approx140.59\)
• \((10 - 29.143)^{2}\approx366.46\)
• \((22 - 29.143)^{2}\approx51.02\)
• \((8 - 29.143)^{2}\approx446.04\)
• \((40 - 29.143)^{2}\approx117.87\)
• \((59 - 29.143)^{2}\approx891.46\)

Sum of \((y_{i}-\bar{y})^{2}\): \(26.45+140.59+366.46+51.02+446.04+117.87+891.46=2039.89\)

Step3: Calculate \(r\)

\(r=\frac{736.34}{\sqrt{266.85\times2039.89}}=\frac{736.34}{\sqrt{544779.66}}=\frac{736.34}{738.1}\approx0.998\)

Answer:

\(0.998\)