Question

find the linear correlation coefficient for stride length versus speed of a camel, as given in the table below. round your result to the nearest thousandth. (see example 3 in this section.)

1. -/12.5 points

find the equation of the least - squares line for the stride length and speed of camels given in the table below. (use x as the independent variable. round all numerical values to two decimal places. see example 1 in this section.)

Explanation:

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

Let \(x = [2.5,3.0,3.2,3.4,3.5,3.8,4.0,4.2]\) and \(y=[2.3,3.9,4.1,5.0,5.5,6.2,7.1,7.6]\)
The mean of \(x\), \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)
\(\sum_{i=1}^{8}x_{i}=2.5 + 3.0+3.2 + 3.4+3.5+3.8+4.0+4.2=27.6\)
\(\bar{x}=\frac{27.6}{8}=3.45\)
The mean of \(y\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\)
\(\sum_{i=1}^{8}y_{i}=2.3 + 3.9+4.1+5.0+5.5+6.2+7.1+7.6=41.7\)
\(\bar{y}=\frac{41.7}{8}=5.2125\)

Step2: Calculate \(n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}\)

\(\sum_{i=1}^{8}x_{i}y_{i}=(2.5\times2.3)+(3.0\times3.9)+(3.2\times4.1)+(3.4\times5.0)+(3.5\times5.5)+(3.8\times6.2)+(4.0\times7.1)+(4.2\times7.6)\)
\(=5.75+11.7+13.12+17+19.25+23.56+28.4+31.92 = 150.6\)
\(n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}=8\times150.6-27.6\times41.7\)
\(=1204.8 - 1151.92=52.88\)

Step3: Calculate \(n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}\)

\(\sum_{i=1}^{8}x_{i}^{2}=2.5^{2}+3.0^{2}+3.2^{2}+3.4^{2}+3.5^{2}+3.8^{2}+4.0^{2}+4.2^{2}\)
\(=6.25 + 9+10.24+11.56+12.25+14.44+16+17.64=97.38\)
\(n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}=8\times97.38-(27.6)^{2}\)
\(=779.04 - 761.76 = 17.28\)

Step4: Calculate \(n\sum_{i = 1}^{n}y_{i}^{2}-(\sum_{i = 1}^{n}y_{i})^{2}\)

\(\sum_{i=1}^{8}y_{i}^{2}=2.3^{2}+3.9^{2}+4.1^{2}+5.0^{2}+5.5^{2}+6.2^{2}+7.1^{2}+7.6^{2}\)
\(=5.29+15.21+16.81+25+30.25+38.44+50.41+57.76=239.37\)
\(n\sum_{i = 1}^{n}y_{i}^{2}-(\sum_{i = 1}^{n}y_{i})^{2}=8\times239.37-(41.7)^{2}\)
\(=1914.96-1738.89 = 176.07\)

Step5: Calculate the slope \(b\)

\(b=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{\sqrt{(n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2})(n\sum_{i = 1}^{n}y_{i}^{2}-(\sum_{i = 1}^{n}y_{i})^{2})}}\)
\(=\frac{52.88}{\sqrt{17.28\times176.07}}=\frac{52.88}{\sqrt{3042.49}}=\frac{52.88}{55.16}\approx0.96\)

Step6: Calculate the y - intercept \(a\)

\(a=\bar{y}-b\bar{x}\)
\(a = 5.2125-0.96\times3.45\)
\(a=5.2125 - 3.312=1.90\)

Answer:

\(\hat{y}=1.90 + 0.96x\)