Question

the table shows the number of flowers in four bouquets and the total cost of each bouquet. what is the correlation coefficient for the data in the table? cost of bouquets number of flowers in the bouquet total cost 8 $12 12 $40 6 $15 20 $20

Explanation:

Step1: Calculate the mean of \(x\) (number of flowers)

\(\bar{x}=\frac{8 + 12+6 + 20}{4}=\frac{46}{4}=11.5\)

Step2: Calculate the mean of \(y\) (total cost)

\(\bar{y}=\frac{12+40 + 15+20}{4}=\frac{87}{4}=21.75\)

Step3: Calculate \(n\sum xy-\sum x\sum y\), \(n\sum x^{2}-(\sum x)^{2}\) and \(n\sum y^{2}-(\sum y)^{2}\)

• \(\sum x = 8+12 + 6+20=46\), \(\sum y=12 + 40+15+20 = 87\)
• \(\sum xy=(8\times12)+(12\times40)+(6\times15)+(20\times20)=96 + 480+90 + 400=1066\)
• \(\sum x^{2}=8^{2}+12^{2}+6^{2}+20^{2}=64 + 144+36 + 400=644\)
• \(\sum y^{2}=12^{2}+40^{2}+15^{2}+20^{2}=144+1600 + 225+400=2369\)

\(n = 4\)

\(n\sum xy-\sum x\sum y=4\times1066-46\times87=4264-4002 = 262\)

\(n\sum x^{2}-(\sum x)^{2}=4\times644-46^{2}=2576 - 2116=460\)

\(n\sum y^{2}-(\sum y)^{2}=4\times2369-87^{2}=9476-7569 = 1907\)

Step4: Calculate the correlation coefficient \(r\)

\(r=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}}=\frac{262}{\sqrt{460\times1907}}\)

\(\sqrt{460\times1907}=\sqrt{877220}\approx936.6\)

\(r=\frac{262}{936.6}\approx0.28\)

Answer:

\(0.28\)