Question

for the following data set, (a) draw a scatter diagram, (b) compute the correlation coefficient, and (c) comment on the type of relation that appears to exist between x and y.

(a) which of the following scatter diagrams corresponds to the given data?

(b) compute the correlation coefficient.
r≈ (round to three decimal places as needed.)

Explanation:

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

The mean of \(x\) values: \(\bar{x}=\frac{2 + 4+7+7+9}{5}=\frac{29}{5} = 5.8\)
The mean of \(y\) values: \(\bar{y}=\frac{14+16+21+22+28}{5}=\frac{101}{5}=20.2\)

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 \(x = 2,y = 14\): \((2 - 5.8)(14 - 20.2)=(- 3.8)\times(-6.2)=23.56\)
• For \(x = 4,y = 16\): \((4 - 5.8)(16 - 20.2)=(-1.8)\times(-4.2)=7.56\)
• For \(x = 7,y = 21\): \((7 - 5.8)(21 - 20.2)=1.2\times0.8 = 0.96\)
• For \(x = 7,y = 22\): \((7 - 5.8)(22 - 20.2)=1.2\times1.8=2.16\)
• For \(x = 9,y = 28\): \((9 - 5.8)(28 - 20.2)=3.2\times7.8 = 24.96\)
• \(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=23.56 + 7.56+0.96+2.16+24.96=59.2\)

• Calculate \((x_{i}-\bar{x})^{2}\):
• For \(x = 2\): \((2 - 5.8)^{2}=(-3.8)^{2}=14.44\)
• For \(x = 4\): \((4 - 5.8)^{2}=(-1.8)^{2}=3.24\)
• For \(x = 7\): \((7 - 5.8)^{2}=1.2^{2}=1.44\)
• For \(x = 7\): \((7 - 5.8)^{2}=1.2^{2}=1.44\)
• For \(x = 9\): \((9 - 5.8)^{2}=3.2^{2}=10.24\)
• \(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=14.44+3.24 + 1.44+1.44+10.24=30.8\)

• Calculate \((y_{i}-\bar{y})^{2}\):
• For \(y = 14\): \((14 - 20.2)^{2}=(-6.2)^{2}=38.44\)
• For \(y = 16\): \((16 - 20.2)^{2}=(-4.2)^{2}=17.64\)
• For \(y = 21\): \((21 - 20.2)^{2}=0.8^{2}=0.64\)
• For \(y = 22\): \((22 - 20.2)^{2}=1.8^{2}=3.24\)
• For \(y = 28\): \((28 - 20.2)^{2}=7.8^{2}=60.84\)
• \(\sum_{i = 1}^{5}(y_{i}-\bar{y})^{2}=38.44+17.64+0.64+3.24+60.84 = 120.8\)

Step3: Calculate the correlation coefficient \(r\)

\(r=\frac{59.2}{\sqrt{30.8\times120.8}}=\frac{59.2}{\sqrt{3710.24}}=\frac{59.2}{60.91}\approx0.972\)

Answer:

\(r\approx0.972\)