Question

a data set is shown below.
{(38,50),(89,80),(75,80),(20,40)}
what is the approximate correlation coefficient when the data is modeled by its linear best - fit equation?
a 0.98
b 0.96
c 0.63
d 0.27

Explanation:

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

Let \(x = [38,89,75,20]\), \(y=[50,80,80,40]\)
\(\bar{x}=\frac{38 + 89+75+20}{4}=\frac{222}{4}=55.5\)
\(\bar{y}=\frac{50 + 80+80+40}{4}=\frac{250}{4}=62.5\)

Step2: Calculate numerator and denominator of correlation coefficient formula

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

\(\sum_{i = 1}^{4}(x_{i}-\bar{x})(y_{i}-\bar{y})=(38 - 55.5)(50 - 62.5)+(89 - 55.5)(80 - 62.5)+(75 - 55.5)(80 - 62.5)+(20 - 55.5)(40 - 62.5)\)
\(=(- 17.5)\times(-12.5)+33.5\times17.5 + 19.5\times17.5+(-35.5)\times(-22.5)\)
\(=218.75+586.25+341.25 + 798.75=1945\)

\(\sum_{i = 1}^{4}(x_{i}-\bar{x})^{2}=(38 - 55.5)^{2}+(89 - 55.5)^{2}+(75 - 55.5)^{2}+(20 - 55.5)^{2}\)
\(=(-17.5)^{2}+33.5^{2}+19.5^{2}+(-35.5)^{2}\)
\(=306.25 + 1122.25+380.25+1260.25 = 3069\)

\(\sum_{i = 1}^{4}(y_{i}-\bar{y})^{2}=(50 - 62.5)^{2}+(80 - 62.5)^{2}+(80 - 62.5)^{2}+(40 - 62.5)^{2}\)
\(=(-12.5)^{2}+17.5^{2}+17.5^{2}+(-22.5)^{2}\)
\(=156.25+306.25+306.25 + 506.25=1275\)

\(r=\frac{1945}{\sqrt{3069\times1275}}\approx\frac{1945}{\sqrt{3913975}}\approx\frac{1945}{1978.3}\approx0.98\)

Answer:

A. \(0.98\)