Question

find the correlation coefficient, r, of the data described below. quincy keeps detailed information about his model airplane collection. the data includes not only the characteristics of each model, but also the time it took to assemble it. he is thinking of buying a new model airplane and decided to look at this information to understand how long assembly might take. from his data, quincy found the number of pieces in each model, x, and how many minutes each took to assemble, y.

pieces	assembly time
44	41
51	36
53	36
92	57

round your answer to the nearest thousandth. r =

Explanation:

Step1: Calculate the means

Let \(x\) be the number of pieces and \(y\) be the assembly - time.
The data points are \((x_1,y_1)=(40,37),(x_2,y_2)=(44,41),(x_3,y_3)=(51,36),(x_4,y_4)=(53,36),(x_5,y_5)=(92,57)\).
The mean of \(x\), \(\bar{x}=\frac{40 + 44+51+53+92}{5}=\frac{280}{5}=56\).
The mean of \(y\), \(\bar{y}=\frac{37 + 41+36+36+57}{5}=\frac{207}{5}=41.4\).

Step2: Calculate the numerator of the correlation - coefficient formula

The formula for the correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}\).
\((x_1-\bar{x})(y_1 - \bar{y})=(40 - 56)(37-41.4)=(-16)\times(-4.4) = 70.4\).
\((x_2-\bar{x})(y_2 - \bar{y})=(44 - 56)(41 - 41.4)=(-12)\times(-0.4)=4.8\).
\((x_3-\bar{x})(y_3 - \bar{y})=(51 - 56)(36 - 41.4)=(-5)\times(-5.4)=27\).
\((x_4-\bar{x})(y_4 - \bar{y})=(53 - 56)(36 - 41.4)=(-3)\times(-5.4)=16.2\).
\((x_5-\bar{x})(y_5 - \bar{y})=(92 - 56)(57 - 41.4)=36\times15.6 = 561.6\).
\(\sum_{i = 1}^{5}(x_i-\bar{x})(y_i - \bar{y})=70.4 + 4.8+27+16.2+561.6=680\).

Step3: Calculate the denominator of the correlation - coefficient formula

\((x_1-\bar{x})^2=(40 - 56)^2=(-16)^2 = 256\).
\((x_2-\bar{x})^2=(44 - 56)^2=(-12)^2 = 144\).
\((x_3-\bar{x})^2=(51 - 56)^2=(-5)^2 = 25\).
\((x_4-\bar{x})^2=(53 - 56)^2=(-3)^2 = 9\).
\((x_5-\bar{x})^2=(92 - 56)^2=36^2 = 1296\).
\(\sum_{i = 1}^{5}(x_i-\bar{x})^2=256+144 + 25+9+1296=1730\).
\((y_1-\bar{y})^2=(37 - 41.4)^2=(-4.4)^2 = 19.36\).
\((y_2-\bar{y})^2=(41 - 41.4)^2=(-0.4)^2 = 0.16\).
\((y_3-\bar{y})^2=(36 - 41.4)^2=(-5.4)^2 = 29.16\).
\((y_4-\bar{y})^2=(36 - 41.4)^2=(-5.4)^2 = 29.16\).
\((y_5-\bar{y})^2=(57 - 41.4)^2=15.6^2 = 243.36\).
\(\sum_{i = 1}^{5}(y_i-\bar{y})^2=19.36+0.16+29.16+29.16+243.36=321.2\).
\(\sqrt{\sum_{i = 1}^{5}(x_i-\bar{x})^2\sum_{i = 1}^{5}(y_i-\bar{y})^2}=\sqrt{1730\times321.2}=\sqrt{555676}\approx745.44\).

Step4: Calculate the correlation coefficient

\(r=\frac{680}{745.44}\approx0.912\).

Answer:

\(0.912\)