Question

use the given data to complete parts (a) and (b).

compute the linear correlation coefficient. the linear correlation coefficient for the four pieces of data is 0.125
(round to three decimal places as needed.)
(b) draw a scatter diagram of the data with the additional data point (10.4,9.4). choose the correct answer.

compute the linear correlation coefficient with the additional data point. the linear correlation coefficient for the five pieces of data is
(round to three decimal places as needed.)

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
For \(n = 5\), \(x\) - values: \(2.1\), \(3.8\), \(3\), \(4.7\), \(10.4\). \(\sum x=2.1 + 3.8+3 + 4.7+10.4 = 24\), \(\bar{x}=\frac{24}{5}=4.8\).
\(y\) - values: \(3.8\), \(1.5\), \(3.6\), \(4.9\), \(9.4\). \(\sum y=3.8 + 1.5+3.6 + 4.9+9.4 = 23.2\), \(\bar{y}=\frac{23.2}{5}=4.64\).

Step2: Calculate the numerator and denominator of the correlation coefficient formula

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

• Numerator:

\((2.1 - 4.8)(3.8 - 4.64)+(3.8 - 4.8)(1.5 - 4.64)+(3 - 4.8)(3.6 - 4.64)+(4.7 - 4.8)(4.9 - 4.64)+(10.4 - 4.8)(9.4 - 4.64)\)
\(=(- 2.7)(-0.84)+(-1)(-3.14)+(-1.8)(-1.04)+(-0.1)(0.26)+(5.6)(4.76)\)
\(=2.268+3.14 + 1.872-0.026+26.656=33.91\)

• Denominator:

\(\sum(x_{i}-\bar{x})^{2}=(2.1 - 4.8)^{2}+(3.8 - 4.8)^{2}+(3 - 4.8)^{2}+(4.7 - 4.8)^{2}+(10.4 - 4.8)^{2}\)
\(=(-2.7)^{2}+(-1)^{2}+(-1.8)^{2}+(-0.1)^{2}+(5.6)^{2}=7.29 + 1+3.24+0.01+31.36 = 42.9\)
\(\sum(y_{i}-\bar{y})^{2}=(3.8 - 4.64)^{2}+(1.5 - 4.64)^{2}+(3.6 - 4.64)^{2}+(4.9 - 4.64)^{2}+(9.4 - 4.64)^{2}\)
\(=(-0.84)^{2}+(-3.14)^{2}+(-1.04)^{2}+(0.26)^{2}+(4.76)^{2}=0.7056+9.8596+1.0816+0.0676+22.6576 = 34.372\)
\(\sqrt{\sum(x_{i}-\bar{x})^{2}\sum(y_{i}-\bar{y})^{2}}=\sqrt{42.9\times34.372}\approx\sqrt{1474.5588}\approx38.4\)

Step3: Calculate \(r\)

\(r=\frac{33.91}{38.4}\approx0.883\)

Answer:

\(0.883\)