Question

question 4
based on the data shown below, calculate the correlation coefficient

x	y
5	26.82
6	26.15
7	23.48
8	22.01
9	21.44
10	18.67
11	20.5

question help:

Explanation:

Step1: Calculate means

Let \(n = 8\).
\(\bar{x}=\frac{4 + 5+6+7+8+9+10+11}{8}=\frac{60}{8}=7.5\)
\(\bar{y}=\frac{27.69+26.82+26.15+23.48+22.01+21.44+18.67+20.5}{8}=\frac{186.76}{8}=23.345\)

Step2: Calculate numerator and denominator components

Let \(S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\), \(S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\), \(S_{yy}=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}\)

For \(S_{xy}\):
\((4 - 7.5)(27.69-23.345)=(- 3.5)\times4.345=-15.2075\)
\((5 - 7.5)(26.82 - 23.345)=(-2.5)\times3.475=-8.6875\)
\((6 - 7.5)(26.15 - 23.345)=(-1.5)\times2.805=-4.2075\)
\((7 - 7.5)(23.48 - 23.345)=(-0.5)\times0.135=-0.0675\)
\((8 - 7.5)(22.01 - 23.345)=0.5\times(-1.335)=-0.6675\)
\((9 - 7.5)(21.44 - 23.345)=1.5\times(-1.905)=-2.8575\)
\((10 - 7.5)(18.67 - 23.345)=2.5\times(-4.675)=-11.6875\)
\((11 - 7.5)(20.5 - 23.345)=3.5\times(-2.845)=-9.9575\)
\(S_{xy}=-15.2075-8.6875 - 4.2075-0.0675-0.6675-2.8575-11.6875-9.9575=-53.33\)

For \(S_{xx}\):
\((4 - 7.5)^{2}=(-3.5)^{2}=12.25\)
\((5 - 7.5)^{2}=(-2.5)^{2}=6.25\)
\((6 - 7.5)^{2}=(-1.5)^{2}=2.25\)
\((7 - 7.5)^{2}=(-0.5)^{2}=0.25\)
\((8 - 7.5)^{2}=0.5^{2}=0.25\)
\((9 - 7.5)^{2}=1.5^{2}=2.25\)
\((10 - 7.5)^{2}=2.5^{2}=6.25\)
\((11 - 7.5)^{2}=3.5^{2}=12.25\)
\(S_{xx}=12.25 + 6.25+2.25+0.25+0.25+2.25+6.25+12.25 = 42\)

For \(S_{yy}\):
\((27.69-23.345)^{2}=4.345^{2}=18.879025\)
\((26.82-23.345)^{2}=3.475^{2}=12.075625\)
\((26.15-23.345)^{2}=2.805^{2}=7.868025\)
\((23.48-23.345)^{2}=0.135^{2}=0.018225\)
\((22.01-23.345)^{2}=(-1.335)^{2}=1.782225\)
\((21.44-23.345)^{2}=(-1.905)^{2}=3.639025\)
\((18.67-23.345)^{2}=(-4.675)^{2}=21.855625\)
\((20.5-23.345)^{2}=(-2.845)^{2}=8.094025\)
\(S_{yy}=18.879025+12.075625+7.868025+0.018225+1.782225+3.639025+21.855625+8.094025 = 74.2118\)

The correlation coefficient \(r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{-53.33}{\sqrt{42\times74.2118}}=\frac{-53.33}{\sqrt{3116.8956}}\approx\frac{-53.33}{55.83}\approx - 0.955\)

Answer:

\(-0.955\)