Question

find the correlation coefficient, r, of the data described below.
the ceo of mcleans natural fruit juice wants to see whether the companys internal tasters give consistent ratings. she set up a blind taste test in which the tasters rated several samples of fruit juice in the same order, not knowing that two of the samples were actually the same pomegranate - kiwi blend.
each taster rated the samples on a 100 - point scale. the ceo recorded the ratings that had been given to the first sample of the pomegranate - kiwi blend, x, and the second, y.
first rating second rating
82 76
85 80
87 81
91 89
91 92
round your answer to the nearest thousandth.
r =

Explanation:

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

Let \(x = [82,85,87,91,91]\) and \(y=[76,80,81,89,92]\).
The mean of \(x\), \(\bar{x}=\frac{82 + 85+87+91+91}{5}=\frac{436}{5}=87.2\)
The mean of \(y\), \(\bar{y}=\frac{76 + 80+81+89+92}{5}=\frac{418}{5}=83.6\)

Step2: Calculate the numerator of the correlation - coefficient formula

\[

$$\begin{align*} \sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(82 - 87.2)(76-83.6)+(85 - 87.2)(80 - 83.6)+(87 - 87.2)(81 - 83.6)+(91 - 87.2)(89 - 83.6)+(91 - 87.2)(92 - 83.6)\\ &=(-5.2)(-7.6)+(-2.2)(-3.6)+(-0.2)(-2.6)+(3.8)(5.4)+(3.8)(8.4)\\ &=39.52+7.92 + 0.52+20.52+31.92\\ &=100.4 \end{align*}$$

\]

Step3: Calculate the denominator of the correlation - coefficient formula

\[

$$\begin{align*} \sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}&=(82 - 87.2)^{2}+(85 - 87.2)^{2}+(87 - 87.2)^{2}+(91 - 87.2)^{2}+(91 - 87.2)^{2}\\ &=(-5.2)^{2}+(-2.2)^{2}+(-0.2)^{2}+(3.8)^{2}+(3.8)^{2}\\ &=27.04+4.84+0.04 + 14.44+14.44\\ &=60.8 \end{align*}$$

\]
\[

$$\begin{align*} \sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}&=(76 - 83.6)^{2}+(80 - 83.6)^{2}+(81 - 83.6)^{2}+(89 - 83.6)^{2}+(92 - 83.6)^{2}\\ &=(-7.6)^{2}+(-3.6)^{2}+(-2.6)^{2}+(5.4)^{2}+(8.4)^{2}\\ &=57.76+12.96+6.76+29.16+70.56\\ &=177.2 \end{align*}$$

\]
The denominator is \(\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}=\sqrt{60.8\times177.2}=\sqrt{10773.76}\approx103.797\)

Step4: Calculate the correlation coefficient \(r\)

\(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{100.4}{103.797}\approx0.967\)

Answer:

\(0.967\)